Homotopy Theory: Tools and Applications: A Conference in Honor of Paul Goerss’s 60th Birthday, July 17-21, 2017, University of Illinois at Urbana-Champaign, Urbana, Illinois 1470442442, 9781470442446

Citation preview

729

Homotopy Theory: Tools and Applications A Conference in Honor of Paul Goerss’s 60th Birthday July 17–21, 2017 University of Illinois at Urbana-Champaign Urbana, Illinois

Daniel G. Davis Hans-Werner Henn J. F. Jardine Mark W. Johnson Charles Rezk Editors

729

Homotopy Theory: Tools and Applications A Conference in Honor of Paul Goerss’s 60th Birthday July 17–21, 2017 University of Illinois at Urbana-Champaign Urbana, Illinois

Daniel G. Davis Hans-Werner Henn J. F. Jardine Mark W. Johnson Charles Rezk Editors

EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss

Kailash Misra

Catherine Yan

2010 Mathematics Subject Classification. Primary 55P43, 55N22, 55N91, 18D50, 55Q45, 55Q51, 55T15, 54B40, 55U10, 55S35.

Library of Congress Cataloging-in-Publication Data Names: Davis, Daniel G., 1972- editor. Title: Homotopy theory : tools and applications : a conference in honor of Paul Goerss’s 60th birthday, July 17-21, 2017, University of Illinois at Urbana-Champaign, Urbana, Illinois / Daniel G. Davis [and four others], editors. Description: Providence, Rhode Island : American Mathematical Society, [2019] | Series: Contemporary mathematics ; volume 729 | Includes bibliographical references. Identifiers: LCCN 2018047360 | ISBN 9781470442446 (alk. paper) Subjects: LCSH: Homotopy theory–Congresses. | Goerss, Paul Gregory, honoree. | Festschriften. | AMS: Algebraic topology – Homotopy theory – Spectra with additional structure ( ring spectra, etc.). msc | Algebraic topology – Homology and cohomology theories – Bordism and cobordism theories, formal group laws. msc | Algebraic topology – Homology and cohomology theories – Equivariant homology and cohomology. msc | Category theory; homological algebra – Categories with structure – Operads. msc | Algebraic topology – Homotopy groups – Stable homotopy of spheres. msc | Algebraic topology – Homotopy groups – periodicity. msc | Algebraic topology – Spectral sequences – Adams spectral sequences. msc | General topology – Basic constructions – Presheaves and sheaves. msc | Algebraic topology – Applied homological algebra and category theory – Simplicial sets and complexes. msc Classification: LCC QA612.7 .H66 2019 | DDC 514/.24–dc23 LC record available at https://lccn.loc.gov/2018047360 Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online) DOI: https://doi.org/10.1090/conm/729

Color graphic policy. Any graphics created in color will be rendered in grayscale for the printed version unless color printing is authorized by the Publisher. In general, color graphics will appear in color in the online version. Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2019 by the American Mathematical Society. All rights reserved.  The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines 

established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

24 23 22 21 20 19

The editors dedicate this volume to Paul Goerss on the occasion of his 60th birthday.

Contents

Preface

vii

Plenary talks

ix

Parallel talks

xi

The α-family in the K(2)-local sphere at the prime 2 Agn` es Beaudry

1

A constructive approach to higher homotopy operations David Blanc, Mark W. Johnson, and James M. Turner

21

The right adjoint to the equivariant operadic forgetful functor on incomplete Tambara functors Andrew J. Blumberg and Michael A. Hill

75

The centralizer resolution of the K(2)-local sphere at the prime 2 Hans-Werner Henn

93

Galois descent criteria J. F. Jardine

129

Quantization of the modular functor and equivariant elliptic cohomology Nitu Kitchloo

157

Calculating obstruction groups for E∞ ring spectra Tyler Lawson

179

Comodules, sheaves, and the exact functor theorem Haynes Miller

205

Complex orientations for THH of some perfectoid fields Jack Morava

221

String bordism and chromatic characteristics Markus Szymik

239

Mahowald square and Adams differentials Zhouli Xu

255

v

Preface This volume of articles is the proceedings for the conference Homotopy Theory: Tools and Applications hosted July 17-21, 2017 by the Department of Mathematics of the University of Illinois at Urbana-Champaign, and organized by Daniel Davis, Mark W. Johnson, Charles Rezk, and Vesna Stojanoska. This conference included over 150 participants representing institutions from a dozen countries, and consisted of 20 one-hour plenary lectures along with 24 half-hour contributed talks organized into four parallel sessions. Lists of these talks appear below. Conference participants were invited to submit original research articles, which all went through a standard refereeing process. The original impetus for organizing the conference was the 60th birthday of Paul Goerss, and the breadth of the articles included here is a reflection of his broad interests within homotopy theory. Paul began his career working with Brown-Gitler spectra. His early Ast´erisque volume on Andre-Quillen cohomology set the stage for his later important work with Mike Hopkins on moduli problems for structured ring spectra, culminating in the Hopkins-Miller Theorem.1 His work on homotopy fixed points foreshadowed his more recent important work with resolutions in chromatic homotopy theory and the chromatic splitting conjecture. In addition, his influential Bourbaki survey on topological modular forms is a fine example of his clear expository style. Paul has also had an impact on how homotopy theory is passed on to the succeeding generation of researchers. In addition to his broad influence through the now standard reference Simplicial Homotopy Theory, he has advised 20 Ph.D. students so far, and he has organized a variety of major events. Among these are 5 Oberwolfach meetings (including a spring 2019 Arbeitsgemeinschaft), 3 Fields Institute meetings, and 2 events at the Newton Institute (including a program throughout the second half of 2018). Now we share a morsel concerning each of the articles included here: • Beaudry: locates the α-family in π∗ (LK(2) S 0 ), for the difficult p = 2 case, thereby aiding the effort to understand π∗ (LK(2) S 0 ) through duality spectral sequences. • Blanc, Johnson, and Turner: provides a very general definition of a higher homotopy operation as an obstruction to rigidifying a homotopy commutative diagram, valid even in unpointed situations. • Blumberg and Hill: details illustrative examples of computations as well as providing an overview of Tambara functors, or the natural structures on the zeroth homotopy groups of algebras over N∞ operads, the truly equivariant versions of E∞ operads. 1 Some

authors refer to this as the Goerss-Hopkins-Miller Theorem, but Paul does not. vii

viii

PREFACE

• Henn: constructs algebraic centralizer resolutions for the Morava stabilizer group G2 and for the related subgroups S12 and G12 , as well as topological realizations of them, thus providing input for recent progress by Beaudry and Bobkova-Goerss in K(2)-local homotopy theory at the prime 2. • Jardine: gives a sheaf theoretic description of Galois descent and homotopy fixed points problems, with a criterion for descent that uses homotopy types of pro-objects, thereby shedding new light on the ´etale K-theory spectrum. • Kitchloo: produces a global version of dominant K-theory, thereby extending some of his work with Morava on S 1 -equivariant K-theory of the free loop space of a manifold in order to incorporate the action of a compact Lie group. • Lawson: shares welcome insight into his recent proof that the BrownPeterson spectrum BP at the prime 2 is not an E∞ ring spectrum, by explaining how he found a certain non-zero Goerss-Hopkins obstruction. • Miller: studies spin formal groups and gives a proof, due to Mike Hopkins, of the Landweber exact functor theorem in all cases of interest, by emphasizing the pivotal role of the height of a formal group. • Morava: contributes to the development of the theory of cyclotomic spectra by considering T HH and an analog of the Chern character for rings of integers in certain perfectoid fields. • Szymik: defines chromatic characteristics associated with the HopkinsMiller classes ζn in π−1 of the K(n)-local sphere, viewing the ζn as generalizations of prime numbers, and provides a variety of interesting examples and non-examples. • Xu: details the RP ∞ -method for computing differentials in the Adams spectral sequence for the sphere spectrum, providing a simpler example of the technique he recently used with Guozhen Wang to solve a difficult extension problem in the 61-stem. Acknowledgements: We would like to thank each of the authors and anonymous referees for their time and effort. We also gratefully acknowledge support for the conference from both the National Science Foundation and the host department, especially from its chairman at the time, Matthew Ando. Daniel G. Davis Hans-Werner Henn J.F. Jardine Mark W. Johnson Charles Rezk

Plenary Talks • • • • • • • • • • • • • • • • • • • •

Agnès Beaudry: K(n)-local Picard Groups and Gross-Hopkins duality Mark Behrens: The tmf resolution for Z David Blanc: Higher order homotopy invariants Anna Marie Bohmann: CoT HH and calculations Hans-Werner Henn: Resolutions in K(2)-local homotopy theory – old and new Kathryn Hess: Configuration spaces of products Mike Hopkins: Equivariant dual of Morava E-theory Marc Hoyois: Motivic infinite loop spaces J.F. Jardine: Galois cohomological descent Magdalena Kędziorek: An algebraic model for rational G-spectra Nitu Kitchloo: Higher Associative structures on Moore Spectra Tyler Lawson: Higher multiplication and the Brown-Peterson spectrum Jacob Lurie: Brauer Groups in Stable Homotopy Theory Haynes Miller: Some homological localization theorems Jack Morava: Chern characters for Lubin-Tate lifts of K(n) Doug Ravenel: The C2 -equivariant analog of the subalgebra of A generated by Sq 1 and Sq 2 Birgit Richter: Juggling formulae for higher T HH Brooke Shipley: Coalgebras, coT HH, and trace maps Zhouli Xu: Computing stable homotopy groups of spheres Inna Zakharevich: Constructing derived zeta functions

ix

Parallel Talks • Lauren Bandklayder: The Dold-Thom Theorem via factorization homology • Kristine Bauer: Higher order chain rules and differential structure for abelian functor calculus • Haldun Ozgur Bayindir: Topological equivalences of E∞ DGAs • Georg Biedermann: A generalized Blakers-Massey Theorem • Jeffrey Carlson: Rational equivariant K-theory of homogeneous spaces • Sunil Chebolu: Strong ghosts in the stable category • Martin Frankland: Eilenberg-MacLane mapping algebras and higher distributivity up to homotopy • Jeremy Hahn: Chromatic types of structured ring spectra • Zhen Huan: Quasi-elliptic cohomology • Brenda Johnson: Functor precalculus • Inbar Klang: Factorization homology and topological Hochschild cohomology of Thom spectra • Ben Knudsen: Subdivisional spaces and graph braid groups • Kathryn Lesh: Tits buildings and fixed points of p-toral groups on decomposition space • Ayelet Lindenstrauss: The topological Hochschild homology of maximal orders in simple algebras over the rationals • Carl McTague: tmf is not a ring spectrum quotient • Tasos Moulinos: On the topological K-theory of derived Azumaya algebras • Sam Nariman: A local to global argument on low dimensional manifolds • Peter Nelson: A small presentation for Morava E-theory power operations • Luis Pereira: Genuine equivariant operads • Piotr Pstragowski: Moduli of Pi-algebras • Nima Rasekh: Representable cartesian fibrations • Jonathan Rubin: On the realization problem for N∞ operads • XiaoLin Danny Shi: Hurewicz images of Real bordism theory and Morava E-theories • Marco Varisco: Assembly maps for topological cyclic homology

xi

Contemporary Mathematics Volume 729, 2019 https://doi.org/10.1090/conm/729/14689

The α-family in the K(2)-local sphere at the prime 2 Agn`es Beaudry Abstract. In this note, we compute the image of the α-family in the homotopy of the K(2)-local sphere at the prime p = 2 by locating its image in the algebraic duality spectral sequence. This is a steppingstone for the computation of the homotopy groups of the K(2)-local sphere at the prime 2 using the duality spectral sequences.

Contents 1. Introduction 2. The α-family in the Adams-Novikov Spectral Sequence 3. Subgroups of G2 and the algebraic duality spectral sequence 4. The α-family in the K(2)-local sphere References

Acknowledgements. This note was born in conversations which happened during the writing of [BGH17] and the author is indebted to her collaborators, Paul Goerss and Hans-Werner Henn. Many of the methods and ideas used here are borrowed from that rich collaboration. She also thanks Zhouli Xu for useful conversations related to this topic. Mark Mahowald knew how the α-family would be detected in the duality spectral sequences and this paper makes his sketches precise. Some of the computations used in the proof of this theorem are also closely related to results of Mahowald and Rezk in [MR09]. 1. Introduction The first periodic family in the homotopy groups of spheres was constructed by Adams in his study of the image of the J homomorphism, which culminated in what is now one of the must-read articles in algebraic topology, On the Groups J(X) – IV [Ada66]. In the last section of this paper, Adams uses self-maps of Moore spaces to construct elements of the homotopy groups of spheres that he denotes by α. These elements are intimately related to K-theory and are part of what is now called the “α-family”. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1725563. c 2019 American Mathematical Society

1

2

A. BEAUDRY

The α-family is one of the few computable families of elements in the stable homotopy groups of spheres. It is the first of its kind and, with its successors the β and γ-families, it now belongs to a collection of classes known as the Greek-letter elements. In their cornerstone paper on periodicity in the Adams-Novikov Spectral Sequence, Miller, Ravenel and Wilson [MRW77] give an intimate connection between the Greek-letter elements and the chromatic spectral sequence, and establish the importance of the chromatic point of view for computations of the homotopy groups of spheres. Chromatic homotopy as it is known today comes from Morava’s insight that there should be higher analogs of p-completed K-theory. They should carry higher Adams operations, and detect periodic families which are generalizations of the image of J. These cohomology theories are called the Morava E-theories En and the associated mod p theories are called the Morava K-theories K(n). The higher Adams operations form a group called the (extended) Morava stabilizer group, denoted Gn . The theories En and K(n) are complex oriented ring spectra whose construction is based on the deformation theory of height n formal groups. The Morava E and K-theories detect periodic families of elements in the homotopy groups of spheres. There are various ways to make this precise. One is through the eyes of Bousfield localization. The Chromatic Convergence Theorem of Hopkins and Ravenel states that the p-local sphere spectrum S(p) is the (homotopy) inverse limit of the Bousfield localizations SEn of the sphere at the Morava E-theories. One then studies S(p) through its images under the natural maps S(p) → SEn . Further, the SEn can be inductively reassembled from the localizations at the Morava K-theories via a homotopy pull-back SEn

/ SK(n)

 SEn−1

 / (SK(n) )En−1 .

These facts highlight the importance of computing both π∗ SEn and π∗ SK(n) . The standard tools for computing these homotopy groups are two closely related spectral sequences. Note that the En -local sphere is equivalent to SE(n) , where E(n) is the Johnson-Wilson spectrum, a “thinner” version of En . The E(n)-AdamsNovikov Spectral Sequence computes the homotopy groups of SE(n)  SEn : Ext∗,∗ E(n)∗ E(n) (E(n)∗ , E(n)∗ ) =⇒ π∗ SE(n) . The second spectral sequence is the K(n)-local En -Adams-Novikov Spectral Sequence, which computes the homotopy groups of SK(n)  EnhGn . Its E2 -term can be identified with continuous cohomology groups: H ∗ (Gn , (En )∗ ) =⇒ π∗ SK(n) . We give an overview of what is known. First SK(0) and SE0 are both the rational sphere SQ . The computation of π∗ SK(1) and π∗ SE1 can be obtained from the classical computations of Adams, Atiyah and others on the image of J and the action of the Adams operations. The computation of π∗ SE2 and π∗ SK(2) are entirely different beasts. Shimomura, Wang and Yabe have done extensive work on computing these homotopy groups at various primes. The case p ≥ 5 is treated

THE α-FAMILY IN THE K(2)-LOCAL SPHERE AT THE PRIME 2

3

in [SY95] and is also nicely presented in [Beh12]. The case p = 3 is treated in [SW02b, Shi00] and the case p = 2 is partially treated in [SW02a, Shi99]. The height two computations are extremely difficult and the answers contain an enormous amount of information that is hard to interpret and analyze. Having multiple points of view seems to have become an imperative for our understanding of chromatic height two phenomena. In [GHMR05], Goerss, Henn, Mahowald and Rezk establish a different approach to height two computations. It relies on resolutions of the K(2)-local sphere called the duality resolutions, from which one obtains various spectral sequences. For certain subgroups G of G2 , the topological duality spectral sequences converge to π∗ E2hG and the algebraic duality spectral sequences converge to H ∗ (G, (E2 )∗ ). The advantage of the duality spectral sequences is that they organize the computations and the answers in a systematic way. For p ≥ 5, these methods are used in [Lad13], for p = 3, in [HKM13] and for p = 2, in [Bea17b] to perform computations for the K(2)-local Moore spectrum. The homotopy of π∗ SK(2) at p = 3 has been analyzed by Goerss, Henn, Karamanov, Mahowald using duality methods, but has not been fully recorded yet. Duality spectral sequence techniques are also being used to solve other central problems in chromatic homotopy theory. They have been crucial in the study of the Chromatic Splitting Conjecture [GHM14, Bea17a, BGH17] at p = 2 and p = 3. In particular, they play a central role in the disproof of the strongest form of the conjecture at p = 2 [Bea17a, BGH17]. The computations of the K(2)-local Picard groups and of the Gross-Hopkins dual of the sphere at the prime p = 3 rely on the duality spectral sequences [GHMR15, GH16]. These are currently being adapted by the author and her collaborators to solve the same problems at p = 2. Finally, Bhattacharya and Egger use the duality techniques to compute the homotopy groups of the first example of a type 2 complex with a v21 self-map [BE17]. The current paper is concerned with computations of π∗ SK(2) at p = 2 using duality spectral sequence techniques and we finish the introduction by stating our result. When computing π∗ SK(2) , a first and essential step is to locate the α-family in the computation. The goal of this paper is to do this at p = 2, using the duality techniques. The results in this paper are a steppingstone for a full computation of π∗ SK(2) using the duality spectral sequences. We will recall the precise definition of the α-family in Section 2. We will define the algebraic duality spectral sequence and the subgroup S12 ⊆ G2 in Section 3. Our main results (in Section 4) are summarized in the following statement. Theorem. Let p = 2. The elements αi/j ∈ Ext1,2i BP∗ BP (BP∗ , BP∗ ) map nontrivially to H 1 (S12 , (E2 )2i ). In the algebraic duality spectral sequence E1p,q,t = H q (Fp , (E2 )t ) =⇒ H p+q (S12 , (E2 )t ) the αs are detected as follows: (a) α2/2 ∈ E10,1,4 (b) αi/1 ∈ E10,1,2i if i ≥ 1 is odd. (c) αi/j ∈ E11,0,2i if i is even. The maps H 1 (G2 , (E2 )t ) → H 1 (S12 , (E2 )t )

4

A. BEAUDRY

in degrees t = 0 are injective so that the image of the αi/j have unique lifts in H 1 (G2 , (E2 )2i ). In the spectral sequence H ∗ (G2 , (E2 )∗ ) =⇒ π∗ SK(2) the α-family supports the standard pattern of differentials and the family of elements hS1 detected by the αs in π∗ S maps non-trivially to π∗ SK(2) . The same holds in π∗ E2 2 and the associated homotopy fixed point spectral sequence. 2. The α-family in the Adams-Novikov Spectral Sequence In this section, we review the construction of the α-family and fix notation. There are many references for these results: See, for example, Section 4 of [MRW77] and Section 4 of [Rav78]. We let S = S(2) and BP be the 2-local Brown-Peterson spectrum. For a spectrum X, the Adams-Novikov Spectral Sequence (ANSS) is given by E2s,t = Exts,t BP∗ BP (BP∗ , BP∗ X) =⇒ πt−s X(2) . The α-family is a collection of elements αi/j ∈ Ext1,∗ BP∗ BP (BP∗ , BP∗ ) which we construct below. Remark 2.1. We also call the collection of non-trivial elements of π∗ S detected by the αs the α-family, or the topological α-family when we wish to make the distinction clear. To define the α-family, one first shows that there is an isomorphism Ext0,∗ (BP∗ , BP∗ /2) ∼ = F2 [v1 ]. BP∗ BP

See for example Theorem 4.3.2 of [Rav86]. The α-family is defined by taking the image in Ext1,∗ BP∗ BP (BP∗ , BP∗ ) of the powers of v1 under various Bockstein homomorphisms. Define x1,n ∈ v1−1 BP2n+1 by ⎧ v1 n=0 ⎪ ⎪ ⎪ ⎨v 2 − 4v −1 v n = 1 2 1 1 x1,n = 4 ⎪ n=2 v1 − 8v1 v2 ⎪ ⎪ ⎩ 2 n ≥ 3. x1,n−1 Let s ≥ 1 be an odd integer. The reduction of xs1,n modulo 2 is an element of n BP2n+1 s /2 congruent to v12 s . Furthermore, ⎧ ⎪ n = 0 and s ≥ 1 ⎨BP2 /2 xs1,n ∈ BP4 /4 n = 1 and s = 1 ⎪ ⎩ n+2 n ≥ 2, or n = 1 and s ≥ 3 BP2n+1 s /2 are comodule primitives. See, for example, Lemma 4.12 of [MRW77]. Let 1,t n δ (n) : Ext0,t BP∗ BP (BP∗ , BP∗ /2 ) → ExtBP∗ BP (BP∗ , BP∗ ) be the connecting Bockstein homomorphism associated to the short exact sequence 0

/ BP∗

×2n

/ BP∗

/ BP∗ /2n

/0.

THE α-FAMILY IN THE K(2)-LOCAL SPHERE AT THE PRIME 2

5

4 2 α1 0  0

α2/2 2

4

α3

α4/4 6

α5

α6/3

α7

α8/5

α9

α10/3

8

10

12

14

16

18

20

8

10

12

14

16

18

20

4 2 η 0  0

ν 2

σ 4

6

Figure 1. The α-family in the E2 (top) and E∞ (bottom) pages of the Adams-Novikov Spectral Sequence. Here, a  denotes a copy of Z2 , a • denotes a copy of Z/2, a • a copy of Z/4 and so on. Dashed lines denote exotic multiplications by 2. Keeping the convention that s ≥ 1 is an odd integer, there are classes αi/j ∈ Ext1,2i BP∗ BP (BP∗ , BP∗ ) of order 2j defined by αs/1 = δ (1) (xs1,0 ), α2/2 = δ (2) (x1,1 ), and α2n s/(n+2) = δ (n+2) (xs1,n ) for n ≥ 2 and s ≥ 1, or for n = 1 and s ≥ 3. We usually abbreviate αi = αi/1 . Note that α1 α2/2 = 0 and otherwise α1k αi/j = 0 for all k ≥ 0. The αi/j s are classes in the E2 -term of the ANSS for the sphere spectrum. This spectral sequence has no d2 differentials for degree reasons, so the E2 and E3 -terms are equal. Further, there are d3 differentials ⎧ 4 ⎪ i=j=1 ⎨α1 d3 (αi/j ) = α13 α4k+1 i = 4k + 3 and j = 1 ⎪ ⎩ 3 α1 α2n s/n+2 i = 2n s + 2 and j = 3. We obtain the pattern in Figure 1, which is also in Table 2 of [Rav78].

6

A. BEAUDRY

3. Subgroups of G2 and the algebraic duality spectral sequence Before turning to the computation of the α-family in the K(2)-local sphere, we recall some of the tools used in the computation. This will be brief, but we refer the reader to [Bea15, Bea17b, BGH17] where these techniques were explained in great detail. We let K(2) refer to the 2-periodic Morava K-theory spectrum whose formal group law is that of the super-singular elliptic curve defined over F4 with Weierstrass equation C 0 : y 2 + y = x3 . The homotopy groups of K(2) are given by K(2)∗ = F4 [u±1 ] for u in degree −2. We let E = E2 be the associated Morava E-theory constructed in [Bea17b, Section 2], chosen so that the formal group law of E is that of the universal deformation of C0 with Weierstrass equation C : y 2 + 3u1 xy + (u31 − 1)y = x3 . Its homotopy groups are E∗ = W[[u1 ]][u±1 ] where u1 is in E0 and u is in E−2 . Here W = W (F4 ) is the ring of Witt vectors on F4 . We choose a primitive third root of unity ω and note that W ∼ = Z2 [ω]/(1 + ω + ω 2 ). This is a complete local ring with residue field F4 . In fact, it is the ring of integers in an unramified extension of degree 2 of Q2 . The Galois group Gal = Gal(F4 /F2 ), whose generator we denote by σ, acts on W by the Z2 -linear map determined by × uller lifts give a natural embedding of F× ω σ = ω 2 . Further, the Teichm¨ 4 ⊆W . We let S2 be the group of automorphisms of the formal group law of K(2). The group S2 is isomorphic to the units in a maximal order O of a division algebra of dimension 4 over Q2 and Hasse invariant 1/2. A presentation for O is given by ∼ WT /(T 2 = −2, aT = T aσ ), a ∈ W. O= It follows that an element of γ ∈ S2 can be written as power series  γ= ai (γ)T i i≥0

where the elements ai (γ) ∈ W satisfy ai (γ)4 − ai (γ) = 0 and a0 (γ) = 0. The group Gal acts on O via its action on W, fixing T . We let G2 be the extension of S2 by Gal, so that G2 = S2  Gal . The right action of S2 on O gives rise to a representation S2 → GL2 (W) whose determinant restricts to a homomorphism det : S2 → Z× 2. We can extend the determinant to G2 by det(x, σ) = det(x). The determinant ∼ composed with the projection to Z× 2 /(±1) = Z2 defines a homomorphism of G2 onto Z2 . For any subgroup G ⊆ G2 , we let G1 be the kernel of this composite. If G is S2 or G2 , this is a split surjection so that ∼ S1  Z2 (3.1) G2 ∼ S2 = = G12  Z2 2

THE α-FAMILY IN THE K(2)-LOCAL SPHERE AT THE PRIME 2

7

We will use the map Z2 → G2 which sends a chosen generator of Z2 to π = 1+2ω ∈ W× as a preferred splitting. The group S2 has the following important subgroups. First, it has a unique conjugacy class of maximal finite subgroups. A representative can be chosen to be the image of the automorphisms of the super-singular curve Aut(C0 ), which we will denote by G24 . It is the semi-direct product of a quaternion group with the natural × copy of F× 4 in S2 . The group C6 = (±1) × F4 is a subgroup of G24 . Note that the 1 torsion is contained in S2 as Z2 is torsion free. So these are in fact subgroups of S12 . However, we note that in S12 the groups G24 and G24 = πG24 π −1 are not conjugate (π ∈ S12 ). Finally, the Galois group acts on these finite subgroups and they can all be extended to corresponding subgroups of G2 . The maximal finite subgroup of G2 is denoted by G48 ∼ = G24  Gal . Next, we turn to the computational tools. For finite spectra X, XK(2)  E hG2 ∧ X  (E ∧ X)hG2 and there is a spectral sequence E2s,t = H s (G2 , Et X) =⇒ πt−s XK(2)

(3.2)

where, here and everywhere, we mean the continuous cohomology groups. Analyzing the E2 -term of this spectral sequence is difficult, so we often start by studying the cohomology of the subgroup S12 . We have an extremely concrete tool to compute the group cohomology of S12 , a spectral sequence called the algebraic duality spectral sequence (ADSS), which we describe here. For a graded profinite Z2 [[S12 ]]-module M (a typical example is M = E∗ X), the algebraic duality spectral sequence for M is a first quadrant spectral sequence: (3.3) E p,q,t = E p,q,t (M ) ∼ = H q (Fp , Mt ) =⇒ H p+q (S1 , Mt ) 1

2

1

with differentials dr : → where F0 = G24 , F1 = F2 = C6 and F3 = G24 . We may omit the internal grading t from the notation. The spectral sequence has an edge homomorphism Erp,q,t

Erp+r,q−r+1,t ,

H p (H 0 (F• , Mt ), d1 ) → H p (S12 , Mt ), where H p (H 0 (F• , Mt ), d1 ) is the cohomology of the complex (3.4)

0

/ H 0 (F0 , Mt )

d1

/ H 0 (F1 , Mt )

d1

/ H 0 (F2 , Mt )

d1

/ H 0 (F3 , Mt )

/ 0,

F and H 0 (Fp , Mt ) = E1p,0,t ∼ = Mt p . Central to the computations of this paper is the differential

d1 : E10,0,t → E11,0,t , which we describe here. There is an element α ∈ W× ⊆ S2 which is defined so that α = 1 + 2ω mod (4) and det(α) = −1.1 It is shown in Theorem 1.1.1 of [Bea17b] that the differential is given by the action of 1 − α: d1 = 1 − α : H 0 (F0 , E∗ ) → H 0 (F1 , E∗ ). 1 At this point, we run into a conflict of notation. In the current trend of K(2)-local computations at p = 2, the element named α plays a crucial and well-established role. We will keep the name, as any element of the α-family has a subscript and this should make it easy to avoid confusion.

8

A. BEAUDRY

To compute with this spectral sequence, we will also need information about the cohomology H ∗ (G24 , E∗ ) (which is isomorphic to H ∗ (G24 , E∗ )) and of H ∗ (C6 , E∗ ). This is all well-known, but nicely presented in Section 2 of [BG18]. So we refer to that paper for the information we need. Finally, for any closed subgroup G of G2 and finite 2-local spectrum X, the complex orientation of Morava E-theory and the fact that the homotopy fixed point spectral sequence (3.2) is isomorphic to the K(2)-local E-based Adams Spectral Sequence (see Appendix A of [DH04]) gives a comparison diagram Ext∗,∗ BP∗ BP (BP∗ , BP∗ X) O ∼ =

+3 π∗ X O ∼ =

Ext∗,∗ M U∗ M U (M U∗ , M U∗ X)

+3 π∗ X

 H ∗ (G2 , E∗ X)

 +3 π∗ XK(2)

 H ∗ (G, E∗ X)

 +3 π∗ (E hG ∧ X).

To detect the α-family in π∗ E hG , one studies the fate of the α-family in Ext∗,∗ BP∗ BP (BP∗ , BP∗ ) under the vertical maps when X = S(2) . 4. The α-family in the K(2)-local sphere We finally turn to the computation of the α-family in the K(2)-local sphere. The approach is as follows. We will identify the image of the α-family under the map ∗ Ext∗,∗ BP∗ BP (BP∗ , BP∗ ) → H (G2 , E∗ ). In particular, we will show that all of the non-trivial classes α1k αi/j map nontrivially. For filtration reasons, this will imply that any class from the topological α-family in π∗ S maps non-trivially to π∗ E hG2 . We will need the following generalization of [BGH17, Proposition 3.2.2], which allows us to identify classes detecting the α-family in the cohomology of certain closed subgroups of Gn . Its proof is completely analogous and is omitted here. Proposition 4.1. Let E = En and H ⊆ Gn be a closed subgroup. Let R = H 0 (H, E0 ). Fix i > 0 and suppose that (1) H 0 (H, E2i /2) is a cyclic R-module generated by v1i . Let yi/j ∈ E2i /2j be a class so that (2) yi/j ≡ v1i modulo 2, and (3) yi/j is invariant under the action of H. Then, up to multiplication by a unit in R, the image of αi/j ∈ π2i−1 E hH is detected in the spectral sequence H s (H, Et ) =⇒ πt−s E hH by the class δ (j) (yi/j ) ∈ H 1 (H, E2i ). One of the consequences of Theorem 1.2.2 of [Bea17b] is the following lemma.

THE α-FAMILY IN THE K(2)-LOCAL SPHERE AT THE PRIME 2

9

Lemma 4.2. Let H be a closed subgroup of G2 which contains S12 . Then H 0 (H, E∗ /2) ∼ = F4 [v1 ]. In particular, any closed subgroup H of G2 which contains S12 satisfies condition (1) of Proposition 4.1 for any i > 0. So, to apply Proposition 4.1, we must identify candidates for the classes yi/j . To construct these classes, recall that there are classical G48 -invariants in E0 associated to the curve C, which play a key role in computations at n = p = 2. Specifically, letting v1 = u1 u−1 and v2 = u−3 the following are invariant for the action of G48 : Δ = 27v2 (v13 − v2 )3 c4 = 9(v14 + 8v1 v2 )   c6 = 27 v16 − 20v13 v2 − 8v22 j = c34 Δ−1 . A few elements in the higher cohomology H ∗ (G48 , E∗ ) will also appear in the computation. Namely, there are elements η ∈ H 1 (G48 , E2 )

ν ∈ H 1 (G48 , E4 )

μ ∈ H 1 (G48 , E6 ).

The classes η and ν are chosen to be the images of α1 and α2/2 under the map ∗ Ext∗,∗ BP∗ BP (BP∗ , BP∗ ) → H (G48 , E∗ ).

We choose the class μ to be the image of α3 . This will be discussed in the proof of Theorem 4.12. It has the property that μ = ηv12 modulo (2) and ηΔ−1 c6 c24 is a unit multiple of jμ. Note that H ∗ (G24 , E∗ ) ∼ = H ∗ (G48 , E∗ ) ⊗Zp W. The restriction H ∗ (G48 , E∗ ) → H ∗ (G24 , E∗ ) is the inclusion of fixed point under the action of the Galois group on the right factor of W. For any element in the cohomology of G48 , we denote its restriction in the cohomology of G24 by the same name. We will prove the following result. Proposition 4.3. Let s ≥ 1 be odd. Then, for the action of G2 , (a) v1s ∈ E2 is an invariant modulo 2, (b) v12 ∈ E4 is an invariant modulo 4, (c) cn4 ∈ E8n is an invariant modulo 2k+4 for n = 2k s where k ≥ 0, and (d) c6 cn4 ∈ E8n+12 is invariant modulo 8 for n ≥ 0. This motivates the following definition, where s ≥ 1 is odd, ⎧ s v1 i = s, j = 1, ⎪ ⎪ ⎪ ⎨v 2 i = j = 2, 1 yi/j = (4.1) k 2 s ⎪ i = 2k+2 s, j = k + 4, c4 ⎪ ⎪ ⎩ (s−3)/2 c6 c4 i = 2s, j = 3, s = 1. Note that in the last two cases of (4.1) (for i = 2k+2 s and j = k + 4, or i = 2s, j = 3, and s = 1), the element yi/j ∈ H 0 (F0 , E2i ) since F0 = G24 ⊆ G48 and both c4 and c6 are invariant for the action of G48 .

10

A. BEAUDRY

Following the outline of Proposition 4.1, we must compute δ (j) (yi/j ). We get specific and do this for the group S12 defined in (3.1) by using the algebraic duality spectral sequence (ADSS) of (3.3). The part of the ADSS relevant for our computations is depicted in Figure 2. Lemma 7.1.2 of [BGH17] gives a method for computing the Bockstein δ (n) of certain elements for the spectral sequence of a double complex which is particularly suited to the ADSS. Combined with Proposition 4.1, it has the following immediate consequence. Theorem 4.4. Let (H 0 (F• , Et ), d1 ) be the complex of (3.4). Let s ≥ 1 be an odd integer. Let (a) i = 2k+2 s and j = k + 4, or (b) i = 2s, j = 3, and s = 1. Then, up to multiplication by a unit in W, αi/j ∈ H 1 (S12 , E2i ) is detected by the image of the class d (y )

1 i/j ∈ H 1 (H 0 (F1 , E2i ), d1 ) 2j under the edge homomorphism p,0,2i ⊆ H p (S12 , E2i ). H 0 (Fp , E2i ) −→ E∞

To prove Proposition 4.3 and thus apply Proposition 4.4, we will need some information about the action of S2 on c4 and c6 which we record now. Proposition 4.5. Let γ = 1 + a2 (γ)T 2 mod T 3 in S2 . Then γ∗ (c4 ) ≡ c4 + 16(a2 (γ) + a2 (γ)2 )v1 v2

mod (32, 16u21 )

γ∗ (c6 ) ≡ c6 + 8(a2 (γ) + a2 (γ)2 )v13 v2

mod (16, 8u41 ).

and Proof. The first claim is Lemma 5.2.2 of [Bea17b]. To prove the second claim, we proceed as in the proof of this lemma. From (3.3.1) of [Bea17b], we have that (4.2)

γ∗ (u) = t0 (γ)u

γ∗ (u1 ) = t0 (γ)u1 +

2 t1 (γ) 3 t0 (γ)

where t0 (γ) ≡ 1 + 2a2 (γ) mod (2, u1 )2 ,

(4.3)

t1 ≡ a2 (γ)2 u1

mod (2, u21 ).

We abbreviate by letting ti = ti (γ) for i = 0, 1 and a2 = a2 (γ). From (4.2), we deduce that, modulo (16) c6 − γ∗ (c6 )

 3 6  u1 t0 + u31 t20 (3t0 + 3u1 t21 + u21 t1 t20 ) + 2(u21 t1 t0 + t60 + 1) . ≡ 4u−6 t−6 0

By Proposition 6.3.3 of [Bea17b], t40 ≡ t0 + u1 t21 + u21 t1 t20

mod (2)

so that c6 − γ∗ (c6 ) ≡ 0 mod (8).

THE α-FAMILY IN THE K(2)-LOCAL SPHERE AT THE PRIME 2

11

To compute the leading term, we consider c6 − γ∗ (c6 ) modulo (16, u41 ). Using (4.3), we have that, modulo (16, u41 )  3 6  u1 (t0 + 3t30 ) + 2u21 t1 t0 + 2(t60 + 1) c6 − γ∗ (c6 ) ≡ 4u−6 t−6 0   ≡ 8u31 u−6 a2 + a22 . In the last line, we used the fact that t60 ≡ 1 mod (2, u41 ) and also modulo (4, u1 ).  Proof of Proposition 4.3. Since v1 is invariant modulo 2, (a) and (b) are immediate. Proposition 4.5 shows that c4 and c6 are invariant under the action of α and π modulo 16 and 8 respectively. Since c4 and c6 are already invariant under the action of G48 and G2 is topologically generated by G48 , α and π, parts (c) and (d) follow by taking appropriate powers.  To apply Proposition 4.4, we will prove something slightly more general: We will completely compute the differential d1 : E10,0,∗ → E11,0,∗ . 1,0,∗ ∼ We first identify E10,0,∗ ∼ = E∗G24 ∼ = H 0 (G24 , E∗ ) and E1 = E∗C6 ∼ = H 0 (C6 , E∗ ) more explicitly than we have done so far. For example, from Section 2 and 3 of [BG18], we have isomorphisms

H 0 (G24 , E∗ ) ∼ = W[[j]][c4 , c6 , Δ±1 ]/(c34 − c26 = (12)3 Δ, c34 = Δj). It follows that the elements n {c6 cm 4 Δ | m ≥ 0, = 0, 1, n ∈ Z}

form a set of topological W-module generators, so that, in the category of profinite graded W-modules,  n H 0 (G24 , E∗ ) ∼ W{c6 cm = 4 Δ }. n,m∈Z,m≥0 =0,1

There is also an isomorphim H 0 (C6 , E∗ ) ∼ = W[[u31 ]][v12 , v1 v2 , v2±2 ]/ ∼ where ∼ is the ideal ((v12 )3 − (v22 )(u31 )2 , (v1 v2 )2 − (v12 )(v22 ), (v12 )(v1 v2 ) − (u31 )(v22 )). Therefore, a basis of topological W-module generators for H 0 (C6 , E∗ ) is given by {(v1 v2 ) (v12 )m (v22 )n | m ≥ 0, = 0, 1, n ∈ Z} and, in the category of profinite graded W-modules,  H 0 (C6 , E∗ ) ∼ W{(v1 v2 ) (v12 )m (v22 )n }. = n,m∈Z,m≥0 =0,1

We are now ready to compute d1 explicitly. We note that this result is intimately related to Propositions 8.1 and 8.2 [MR09].

12

A. BEAUDRY

Proposition 4.6. The differential d1 : E10,0 → E11,0 is determined by the following information: (a) For n, m ∈ Z of the form n = 2k (2t + 1), m ≥ 0 and for = 0, 1, k

n 2 3·2 d1 (c6 cm 4 Δ ) ≡ (v1 )

+2m+3

k

(v22 )2

k

mod (2, v19·2

(1+4t)

+4m+6

)

0

and d1 (Δ ) = 0. (b) For n ∈ Z of the form n = 2k (2t + 1), n ≥ 1, 4(n−1)+2

d1 (cn4 ) ≡ 2k+4 (v1 v2 )(v12 )2(n−1)

mod (2k+5 , 2k+4 v1

).

(c) For n ∈ Z, n ≥ 1 of the form n = 2k (2t + 1) or for n = 0, d1 (c6 cn4 ) ≡ 8(v1 v2 )(v12 )2n+1

mod (16, 8v14n+4 ).

Proof. This differential is given by the action of 1 − α. Further, α ≡ 1 + ωT 2 modulo T 4 for ω a primitive third root of unity. Hence, a2 (α) + a2 (α)2 = −1. The claim (a) is an immediate consequence of Proposition 5.1.1 of [Bea17b], which states that k

2k+1 (4t+1)

α∗ (Δn ) ≡ Δn + v16·2 v2

k

mod (2, u9·2 ), 1

using the fact that c4 ≡ v14 and c6 ≡ v16 modulo 2. To prove (b), from Proposition 4.5, using the fact that c4 ≡ v14 modulo 2, we deduce that k

4(2k −1)

k

α∗ (c24 ) ≡ (c24 + 2k+4 v1

4(2k −1)+2

mod (2k+5 , 2k+4 u1

v1 v2 )

).

Hence, 4(n−1)

α∗ (cn4 ) ≡ cn4 + 2k+4 v1

4(n−1)+2

mod (2k+5 , 2k+4 u1

v1 v2

).

Similarly, to prove (c), using that α∗ (c4 ) ≡ c4 modulo (16) we have α∗ (c6 cn4 ) ≡ α∗ (c6 )cn4

mod (16)

≡ c6 cn4 + 8v14n+3 v2

mod (16, 8v14n+4 ).



Remark 4.7. For ( , a, b) such that = 0, 1, a ≥ 0, and b ∈ Z, we define elements b,a,b in E11,0,t ∼ = H 0 (C6 , Et ) for t = 8 + 4a + 12b that satisfy b,a,b = (v1 v2 ) (v12 )a (v22 )b + . . . as follows: (a) For n, m ∈ Z of the form n = 2k (2t + 1), m ≥ 0 and for = 0, 1, n b0,3·2k +2m+3,2k (1+4t) = d1 (c6 cm 4 Δ )

(b) For n ∈ Z of the form n = 2k (2t + 1), n ≥ 1, d1 (cn4 ) . 2k+4 (c) For n ∈ Z, n ≥ 1 of the form n = 2k (2t + 1) or for n = 0, b1,2(n−1),0 =

b1,2n+1,0 =

d1 (c6 cn4 ) . 8

(d) In all other cases, b,a,b = (v1 v2 ) (v12 )a (v22 )b .

THE α-FAMILY IN THE K(2)-LOCAL SPHERE AT THE PRIME 2

13

Although we will not refer to all of the elements b,a,b defined above, it will be useful to have a fixed name for them in future computations. We now give some consequences of Proposition 4.6. We start with an immediate corollary: Theorem 4.8. In the ADSS E p,q,t ∼ = H q (Fp , Et ) =⇒ H p+q (S1 , Et ) 2

1

there is an isomorphism 0,0,0 ∼ E20,0,0 ∼ = E∞ = W{Δ0 } 0,0,t where Δ0 is the unit in E10,0,0 ∼ = 0 if t = 0. For = H 0 (G24 , E0 ). Further, E2 1,0,t 2,0,t r ≥ 0, the classes b1,r,0 are in the kernel of d1 : E1 → E1 and detect classes 1,0,t in E∞ of degree t = 8 + 4r. These classes have order 8 if r = 2n + 1 and n ≥ 0. They have order 2k+4 if r = 2n for n = 2k s − 1, s ≥ 1 odd, and k ≥ 0.

Remark 4.9. Since the edge homomorphism of the ADSS has the form p,0,∗ H 0 (Fp , E∗ ) −→ E∞ ⊆ H p (S12 , E∗ ), p,0,∗ even if the generators b1,n,0 are strictly speaking elements of E∞ , they repre1 sent unique elements in the cohomology of S2 , and hence, we can write b1,n,0 ∈ H ∗ (S12 , E∗ ) without any ambiguity.

As an immediate consequence of Theorem 4.8, we have the following result, which was already proved in [BGH17]: Corollary 4.10. The inclusion Z2 → E0 induces an isomorphism H 0 (S12 , E∗ ) ∼ = H 0 (S12 , E0 ) ∼ =W 0 1 0 1 ∼ H (G , E∗ ) = H (G , E0 ) ∼ = Z2 2

2

H (G2 , E∗ ) ∼ = H 0 (G2 , E0 ) ∼ = Z2 . 0

Proof. Theorem 4.8 implies that H 0 (S12 , E∗ ) ∼ = W. The result follows for G12 0 1 0 1 ∼ since H (S2 , E∗ ) = H (G2 , E∗ ) ⊗Z2 W with the natural action of Gal on W (see [BG18, Lemma 1.24]). The fixed points for G2 include in those for G12 and contain  the image of Z2 ⊆ E0 . The next three results are depicted in Figure 3. Corollary 4.11. Up to multiplication by a unit in W, (a) b1,2n+1,0 for n ≥ 0 detects α(4n+6)/3 , and (b) b1,2n,0 for n = 2k s − 1, s ≥ 1 odd, k ≥ 0 detects α2k+2 s/(k+4) . Proof. This follows from Proposition 4.4, using Theorem 4.8 and Corollary 4.10.  We turn to the elements αs = αs/1 where s ≥ 1 is odd. Theorem 4.12. Let s ≥ 1 be an odd integer. In the ADSS E p,q,t ∼ = H q (Fp , Et ) =⇒ H p+q (S1 , Et ) 1

2

14

A. BEAUDRY

there are isomorphisms E20,1,2s

∼ =



0,1,2s E∞

∼ =

F4 {ηc6 cm 4 } F4 {μ}

s = 3, s = 1 + 6 + 4m s = 3.

Further αs = αs/1 ∈ H 1 (S12 , E2s ) is non-trivial. The edge homomorphisms 0,1,2s H 1 (S12 , E2s ) → E∞ ⊆ H 1 (F0 , E2s )

are isomorphisms and αs can be identified with its image in H 1 (F0 , E2s ). The element α1 is detected by η, α3 is detected by μ. If s ≥ 5, then s = 1 + 6 + 4m for some = 0, 1 and m ≥ 0. In this case, αs is detected by ηc6 cm 4 . 0,1,2s Proof. The associated graded of the ADSS for H 1 (S1 , E2s ) consists of E∞ 1,0,2s ∼ C6 1,0,2s and E∞ . The latter is a subquotient of E2 = E2s , which is trivial when 0,1,2s s is odd. Therefore, H 1 (S12 , E2s ) ∼ and the edge homomorphism is an = E∞ isomorphism. Now, note that the reduction modulo 2 induces isomorphisms

H 1 (Fp , E2s ) ∼ = H 1 (Fp , E2s /2) 6+4m . So, to compute for p = 0, 1. Further, this isomorphism maps ηc6 cm 4 to ηv1 0,1,2s we can use the commutative diagram E∞

/ H 1 (F0 , E2s )

0

∼ =

d1

∼ =



/ H 1 (F0 , E2s /2)

0

/ H 1 (F1 , E2s )

d1

 / H 1 (F1 , E2s /2)

The kernel of d1 for the top row is isomorphic to the kernel of d1 for the bottom row, which was computed in [Bea17b] to be generated by ηv1s−1 = ηv16+4m if s = 1 + 6 + 4m. Therefore, E20,1,2s ∼ = F4 {ηc6 cm 4 } as desired. Also implied by the extensive computations in [Bea17b] is the fact that H 0 (S12 , E∗ ) ∼ = F4 [v1 ] with the edge homomorphism ∼ =

H 0 (S12 , E2s /2)

0,0,2s / E∞ ⊆ H 0 (F0 , E2s /2)

an isomorphism. Consider the commutative diagram δS1 2

H 0 (S12 , E2s /2)

/ H 1 (S12 , E2s ) ∼ =

∼ =





0,0,2s E∞



0,1,2s E∞

⊆

H 0 (F0 , E2s /2)

δF0



⊆

/ H 1 (F0 , E2s )

where δG is the connecting homomorphism for the exact sequence 0

/ E∗

×2

/ E∗

/ E∗ /2

/ 0.

THE α-FAMILY IN THE K(2)-LOCAL SPHERE AT THE PRIME 2

Since δF0 (v1s )

ηc6 cm 4 = μ

15

s ≥ 1, s = 3, s = 1 + 6 + 4m s = 3,

the image of αs is ηc6 cm 4 if s = 3 and μ if s = 3. So, the corresponding elements of 0,1,2s 0,1,2s ∼ , generated by the E20,1,2s are permanent cycles in the ADSS. So E∞ = E2  image of αs for s odd. It remains to understand the image of α2/2 . Theorem 4.13. In the ADSS, there is an isomorphism 0,1,4 ∼ E20,1,4 ∼ = E∞ = W/4{ν}.

The edge homomorphism 0,1,4 H 1 (S12 , E4 ) → E∞ ⊆ H 1 (F0 , E4 )

is an isomorphism and the element α2/2 can be identified with its image in H 1 (F0 , E4 ), where it is detected by ν. 0,1,4 and Proof. The contributions to H 1 (S12 , E4 ) in the ADSS consist of E∞ There is an isomorphism

1,0,4 . E∞

E10,1,4 ∼ = H 1 (F0 , E4 ) ∼ = H 1 (G24 , E4 ) ∼ = W/4{ν} and ν is so named because it is the image of α2/2 under the homomorphism from ∗ 1 the ANSS E2 -term Ext1,4 BP∗ BP (BP∗ , BP∗ ). This map factors through H (S2 , E∗ ), 0,1,4 so ν must be a permanent cycle in the ADSS. So, all elements of E1 persist to E∞ . 1,0,4 and show that We turn our attention to E∞ 1,0,4 E∞ (E∗ ) = E21,0,4 (E∗ ) ∼ = H 1 (H 0 (F• , E4 )) = 0.

This implies that the edge homomorphism is an isomorphism, and that ν lifts uniquely to an element of H 1 (S12 , E4 ) where it corresponds to the image of α2/2 . We begin with a computation modulo (2). There is a commutative diagram: H 0 (F0 , E4 /2) O

d1

v12 ∼ =

/ H 0 (F1 , E4 /2) O

d1

v12 ∼ =

/ H 0 (F2 , E4 /2) O

d1

v12 ∼ =

/ H 0 (F3 , E4 /2) O v12 ∼ =

H 0 (F0 , E0 /2) O

d1

/ H 0 (F1 , E0 /2) O

d1

/ H 0 (F2 , E0 /2) O

d1

/ H 0 (F3 , E0 /2) O

H 0 (F0 , W/2)

d1

/ H 0 (F1 , W/2)

d1

/ H 0 (F2 , W/2)

d1

/ H 0 (F3 , W/2)

By Theorem 5.4.1 of [BGH17], the vertical map from the bottom to the middle row induces an isomorphism upon taking cohomology with respect to d1 . The cohomology of the bottom row gives a copy of F4 in each degree, whose generators were called Δ0 , b0 , b0 , Δ0 for p = 0, 1, 2, 3 respectively. It follows that F4 {v12 Δ0 } p = 0 p,0,4 ∼ E2 (E∗ /2) = F4 {v12 b0 } p = 1.

16

A. BEAUDRY

d1

1 ) that of Let ker(d1 ) be the kernel of H 0 (F1 , E4 ) −→ H 0 (F2 , E4 ) and ker(d d

1 H 0 (F2 , E4 /2). The diagram H 0 (F1 , E4 /2) −→

H 0 (F0 , E4 /2)

∼ =

/ F4 {v12 Δ0 } ⊕ H 0 (F0 , E4 )/2

d1



ker(d1 )

∼ =



0⊕d1

/ F4 {v12 b0 } ⊕ ker(d1 )/(2)

commutes. Given the cohomology of the left vertical map, it must be the case that d1 induces an isomorphism ∼ =

→ ker(d1 )/(2). d1 : H 0 (F0 , E4 )/2 − So, we have a commutative diagram (4.4)

H 0 (F0 , E4 ) 

d1

/ ker(d1 )

d1

 / ker(d1 )

 / H 1 (H 0 (F• , E4 ))

 / ker(d1 )/(2)

 / 0.

×2

/ H 1 (H 0 (F• , E4 )) ×2

×2

H 0 (F0 , E4 )  H 0 (F0 , E4 )/2

d1 ∼ =

The left two columns of (4.4) are short exact by definition. By Theorem 4.8 E20,0,4 = 0, so the map d1 : H 0 (F0 , E4 ) → H 0 (F1 , E4 ) is injective. So the rows of (4.4) are short exact. It follows that the third column is short exact. So multiplication by 2 is an isomorphism on H 1 (H 0 (F• , E4 )). Since this is a complete Z2 -module, it must be trivial.  We now identify the α-family in H ∗ (G2 , E∗ ). We begin with an observation. Remark 4.14. Recall once more that H ∗ (S12 , E∗ ) ∼ = H ∗ (G12 , E∗ ) ⊗Z2 W and that the restriction H ∗ (G12 , E∗ ) ∼ = H ∗ (S12 , E∗ )Gal

⊆

/ H ∗ (S12 , E∗ )

is an inclusion. Further, the map from the E2 -term of the ANSS to H ∗ (S12 , E∗ ) factors through this inclusion. We have identified the W-submodule W/2j {αi/j } ⊆ H 1 (S12 , E2i ). Choose a Galois invariant W-module generator of W/2j {αi/j } and call it αi/j . Then Z/2j {αi/j } ⊆ H ∗ (G12 , E∗ ). On the other hand, the restriction res : H ∗ (G2 , E∗ ) → H ∗ (G12 , E∗ ) is not injective. So one must proceed with care. Recall that there is a split exact sequence / G2 / Z× /(±1) / G12 / 1. 1 2 ∼ ∼ 1 Using the fact that Z× 2 /(±1) = Z2 , we have G2 = G2  Z2 . We choose π to be 1 a topological generator for Z2 ∼ = G2 /G2 , which acts on H ∗ (G12 , E∗ ). From the

THE α-FAMILY IN THE K(2)-LOCAL SPHERE AT THE PRIME 2

1

v12

η Δ−1 c6 c4 −4 −2

1 0

ν 2

v1 v2

μ Δ−1 c6 c24 4 6

 c4 8

v22

10

v12 v22

c6 12

14

(v1 v2 )v22

c24 16

17

(v22 )2

c 6 c4 20 22

18

Δ 24

Figure 2. A part of the E1 -term of the algebraic duality spectral sequence, E1p,q,t = H q (Fp , Et ). The top is E11,q,t for 0 ≤ q ≤ 3, drawn in the (t − q − 1, q)-plane. The bottom is E10,q,t in the same range, drawn in the (t − q, q)-plane. A  denotes a copy of W[[j]] if p = 0 and W[[u31 ]] if p = 1. A ◦ denotes a copy of F4 [[j]] if p = 0 and F4 [[u31 ]] if p = 1. A • is a copy of F4 and • a copy of W/4. The labels denote the generators as W[[j]]-modules on the p = 0-line and as W[[u31 ]]-modules on the p = 1-line. The lines denote multiplication by η and ν. The dashed line indicates that ηΔ−1 c6 c24 = jμ. Lyndon-Hochschild-Serre Spectral Sequence for the group extension, one obtains a long exact sequence (4.5)

...

/ H ∗ (G12 , E∗ )

π−1

/ H ∗ (G12 , E∗ )

δ

/

H ∗+1 (G2 , E∗ )

res

/ H ∗+1 (G12 , E∗ )

π−1

/ ...

To fully analyze the long exact sequence (4.5), we would need a full computation of H ∗ (G12 , E∗ ), and control over the action of π on the cohomology groups H ∗ (G12 , E∗ ). Neither is available to us at this point. However, in the range of interest for computing the α-family, we get lucky. Proposition 4.15. The restriction H 1 (G2 , Et ) → H 1 (G12 , Et ) is injective if t = 0. Proof. By Corollary 4.10, H 0 (G12 , E∗ ) ∼ = Z2 and the restriction H 0 (G2 , E∗ ) → 1 H (G2 , E∗ ) is an isomorphism. This is the first map in (4.5), so we get an exact sequence 0

0

/ H 0 (G12 , E∗ )

δ

/ H 1 (G2 , E∗ )

/ H 1 (G12 , E∗ )

The claim follows from the fact that H 0 (G12 , Et ) = 0 if t = 0.

π−1

/ H 1 (G12 , E∗ ) 

Corollary 4.16. There are unique classes αi/j ∈ H 1 (G2 , E2i ) which map to the same named classes in H 1 (S12 , E2i ) as described in Remark 4.14. These are the images of the α-family elements under the map from the E2 -term of the BP -based Adams-Novikov Spectral Sequence.

18

A. BEAUDRY

α4/4

α2/2

α1 −4

−2

 0

2

4

α3

α6/3

α5 6

8

α8/5

α7 10

12

α10/3

α9 14

16

α12/4

α11/1 18

20

22

24

Figure 3. The contribution to the α-family in the algebraic duality spectral sequence. The differentials indicate differentials that occur in the homotopy fixed points spectral sequence 1 H ∗ (S12 , E∗ ) =⇒ π∗ E hS2 and the dashed arrows indicate exotic extensions on the E∞ -term of that spectral sequence. Corollary 4.17. The topological α-family maps non-trivially in π∗ SK(2) , and 1 in π∗ E hS2 . The α-family is detected in H ∗ (G2 , E∗ ) by the classes α1k αi/j which support the standard pattern of differentials in the spectral sequence (4.6)

H s (G2 , Et ) =⇒ πt−s E hG2 .

Proof. The only thing to justify is that there are no differentials killing nontrivial elements of the image of the topological α-family. However, the ANSS filtration of the α-elements detecting non-trivial elements in homotopy is at most 3. The first differential in (4.6) and the analogue for S12 is a d3 , and the zero line of (4.6) consists of the permanent cycles H 0 (G2 , E∗ ) ∼ = Z2 , respectively H 0 (S12 , E∗ ) ∼ = W. That the latter are all permanent cycles follows from Section 1.2 of [BG18].  References [Ada66] [BE17] [Bea15] [Bea17a] [Bea17b] [Beh12] [BG18] [BGH17] [DH04]

[GH16]

J. F. Adams, On the groups J(X). IV, Topology 5 (1966), 21–71, DOI 10.1016/00409383(66)90004-8. MR0198470 Prasit Bhattacharya and Philip Egger, Towards the K(2)-local homotopy groups of Z, ArXiv e-prints (2017). Agn` es Beaudry, The algebraic duality resolution at p = 2, Algebr. Geom. Topol. 15 (2015), no. 6, 3653–3705, DOI 10.2140/agt.2015.15.3653. MR3450774 Agn` es Beaudry, The chromatic splitting conjecture at n = p = 2, Geom. Topol. 21 (2017), no. 6, 3213–3230, DOI 10.2140/gt.2017.21.3213. MR3692966 Agn` es Beaudry, Towards the homotopy of the K(2)-local Moore spectrum at p = 2, Adv. Math. 306 (2017), 722–788, DOI 10.1016/j.aim.2016.10.020. MR3581316 Mark Behrens, The homotopy groups of SE(2) at p ≥ 5 revisited, Adv. Math. 230 (2012), no. 2, 458–492, DOI 10.1016/j.aim.2012.02.023. MR2914955 I. Bobkova and P. G. Goerss, Topological resolutions in K(2)-local homotopy theory at the prime 2, J. Topol. 11 (2018), no. 4, 917–956. Agn` es Beaudry, Paul G. Goerss, and Hans-Werner Henn, Chromatic splitting for the K(2)-local sphere at p = 2, ArXiv e-prints (2017). Ethan S. Devinatz and Michael J. Hopkins, Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups, Topology 43 (2004), no. 1, 1–47, DOI 10.1016/S0040-9383(03)00029-6. MR2030586 Paul G. Goerss and Hans-Werner Henn, The Brown-Comenetz dual of the K(2)-local sphere at the prime 3, Adv. Math. 288 (2016), 648–678, DOI 10.1016/j.aim.2015.08.024. MR3436395

THE α-FAMILY IN THE K(2)-LOCAL SPHERE AT THE PRIME 2

19

Paul G. Goerss, Hans-Werner Henn, and Mark Mahowald, The rational homotopy of the K(2)-local sphere and the chromatic splitting conjecture for the prime 3 and level 2, Doc. Math. 19 (2014), 1271–1290. MR3312144 [GHMR05] P. Goerss, H.-W. Henn, M. Mahowald, and C. Rezk, A resolution of the K(2)-local sphere at the prime 3, Ann. of Math. (2) 162 (2005), no. 2, 777–822, DOI 10.4007/annals.2005.162.777. MR2183282 [GHMR15] Paul Goerss, Hans-Werner Henn, Mark Mahowald, and Charles Rezk, On Hopkins’ Picard groups for the prime 3 and chromatic level 2, J. Topol. 8 (2015), no. 1, 267–294, DOI 10.1112/jtopol/jtu024. MR3335255 [HKM13] Hans-Werner Henn, Nasko Karamanov, and Mark Mahowald, The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited, Math. Z. 275 (2013), no. 3-4, 953–1004, DOI 10.1007/s00209-013-1167-4. MR3127044 [Lad13] Olivier Lader, Une r´ esolution projective pour le second groupe de Morava pour p ≥ 5 et applications, Ph.D. thesis, University of Strasbourg, 2013. [MR09] Mark Mahowald and Charles Rezk, Topological modular forms of level 3, Pure Appl. Math. Q. 5 (2009), no. 2, Special Issue: In honor of Friedrich Hirzebruch., 853–872, DOI 10.4310/PAMQ.2009.v5.n2.a9. MR2508904 [MRW77] Haynes R. Miller, Douglas C. Ravenel, and W. Stephen Wilson, Periodic phenomena in the Adams-Novikov spectral sequence, Ann. of Math. (2) 106 (1977), no. 3, 469–516, DOI 10.2307/1971064. MR0458423 [Rav78] Douglas C. Ravenel, A novice’s guide to the Adams-Novikov spectral sequence, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. 404–475. MR513586 [Rav86] Douglas C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics, vol. 121, Academic Press, Inc., Orlando, FL, 1986. MR860042 [Shi99] Katsumi Shimomura, The Adams-Novikov E2 -term for computing π∗ (L2 V (0)) at the prime 2, Topology Appl. 96 (1999), no. 2, 133–152, DOI 10.1016/S01668641(98)00048-0. MR1702307 [Shi00] Katsumi Shimomura, The homotopy groups of the L2 -localized mod 3 Moore spectrum, J. Math. Soc. Japan 52 (2000), no. 1, 65–90, DOI 10.2969/jmsj/05210065. MR1727130 [SW02a] Katsumi Shimomura and Xiangjun Wang, The Adams-Novikov E2 -term for π∗ (L2 S 0 ) at the prime 2, Math. Z. 241 (2002), no. 2, 271–311, DOI 10.1007/s002090200415. MR1935487 [SW02b] Katsumi Shimomura and Xiangjun Wang, The homotopy groups π∗ (L2 S 0 ) at the prime 3, Topology 41 (2002), no. 6, 1183–1198, DOI 10.1016/S0040-9383(01)00033-7. MR1923218 [SY95] Katsumi Shimomura and Atsuko Yabe, The homotopy groups π∗ (L2 S 0 ), Topology 34 (1995), no. 2, 261–289, DOI 10.1016/0040-9383(94)00032-G. MR1318877 [GHM14]

Department of Mathematics, University of Colorado Boulder, Campus Box 395, Boulder, Colorado 80309

Contemporary Mathematics Volume 729, 2019 https://doi.org/10.1090/conm/729/14690

A constructive approach to higher homotopy operations David Blanc, Mark W. Johnson, and James M. Turner To Paul Goerss, on his 60th birthday Abstract. In this paper we provide an explicit general construction of higher homotopy operations in model categories, which include classical examples such as (long) Toda brackets and (iterated) Massey products, but also cover unpointed operations not usually considered in this context. We show how such operations, thought of as obstructions to rectifying a homotopy-commutative diagram, can be defined in terms of a double induction, yielding intermediate obstructions as well.

Introduction Secondary homotopy and cohomology operations have always played an important role in classical homotopy theory (see, e.g., [Ada, BJM, MP, PS] and later [P1, P2, Ald, MO, Sn, CW]), as well as other areas of mathematics (see [AlS, FGM, GL, Gr, SS]). Toda’s construction of what we now call Toda brackets in [T1] (cf. [T2, Ch. I]) was the first example of a secondary homotopy operation stricto sensu, although Adem’s secondary cohomology operations and Massey’s triple products in cohomology appeared at about the same time (see [Ade, Mas]). In [Ada, Ch. 3], Adams first tried to give a general definition of secondary stable cohomology operations (see also [Ha]). Kristensen gave a description of such operations in terms of chain complexes (cf. [Kri,KK]), which was extended by Maunder and others to n-th order cohomology operations (see [Mau,Hol,K1,K2]). Higher operations have also figured over the years in rational homotopy theory, where they are more accessible to computation (see, e.g., [Ald, Basu, Re, Ta]). In more recent years there has been a certain revival of interest in the subject, notably in algebraic contexts (see for example, [Bask, Ga, Sa, E, CF, HW]). In [Sp2], Spanier gave a general theory of higher order homotopy operations (extending the definition of secondary operations given in [Sp1]). Special cases of higher order homotopy operations appeared in [W, Kra, Mo, BBG, OO], and other general definitions may be found in [BM, BJT2].

2010 Mathematics Subject Classification. Primary 55P99; Secondary 18G55, 55Q35, 55S20. Key words and phrases. Higher homotopy operations, homotopy-commutative diagram, obstructions. c 2019 American Mathematical Society

21

22

D. BLANC, M. W. JOHNSON, AND J. M. TURNER

The last two approaches cited present higher order operations as the (last) obstruction to rectifying certain homotopy-commutative diagrams (in spaces or other model categories). In particular, they highlight the special role played by null maps in almost all examples occurring in practice. Implicitly, they both assume an inductive approach to rectifying such diagrams. However, in earlier work no attempt was made to describe a useable inductive procedure, which should (inter alia) explain precisely which lower-order operations are required to vanish in order for a higher order operation to be even defined. The goal of the present note is to make explicit the inductive process underlying our earlier definitions of higher order operations, in as general a framework as possible. We hope the explicit nature of this approach will help in future work both to clarify the question of indeterminacy of the higher operations, and possibly to produce an “algebra of higher operations,” in the spirit of Toda’s original “juggling lemmas” (see [T2, Ch. I]). An important feature of the current approach is that we assume that our indexing category is directed, and we consistently proceed in one direction in rectifying the given homotopy-commutative diagram (say, from right to left, in the “right justified” version). As a result, when we come to define the operation associated to an indexing category of length n, we use as initial data a specific choice of rectification for the right segment of length n − 1. This sequence of earlier choices will appear only implicitly in our description and general notation for higher operations, but will be made explicit for our (long) Toda brackets (see §1.7-4.9). Since our higher operations appear as obstructions to rectification, they fit into the usual framework of obstruction theory: when they do not vanish, one must go back along the thread of earlier choices until reaching a point from which one can proceed along a new branch. From the point of view of the obstruction theory, the important fact is their vanishing or non-vanishing (see Remark 4.9 for the relation to coherent vanishing). Nevertheless, since our higher operations are always described as a certain set of homotopy classes of maps into a suitable pullback, at least in some cases it is possible to describe the indeterminacy more explicitly. However, this would only be a part of the total indeterminacy, since the most general obstruction to rectification consists of the union of these sets, taken over all possible choices of initial data of length n − 1. After a brief discussion of the classical Toda bracket from our point of view in Section 1, in Section 2.A we describe the basic constructions we need, associated to the type of Reedy indexing categories for the diagrams we consider. The changes needed for pointed diagrams are discussed in Section 2.B. We give our general definition of higher order operations in Section 3: it is hard to relate this construction to more familiar examples, because it is intended to cover a number of different situations, and in particular the less common unpointed version. In all cases the “total higher operation” serves as an obstruction to extending a partial rectification of a homotopy-commutative diagram one further stage in the induction. In Section 4 we provide a refinement of this obstruction to a sequence of intermediate steps (in an inner induction), culminating in the total operation for the given stage in the induction. Section 5 is devoted to a commonly occurring problem: rigidifying a (reduced) simplicial object in a model category, for which the simplicial identities hold only up to homotopy. This serves to illustrate how the general (unpointed) theory works in low dimensions.

A CONSTRUCTIVE APPROACH TO HIGHER HOMOTOPY OPERATIONS

23

In Section 6 we define pointed higher operations, which arise when the indexing category has designated null maps, and we want to rectify our diagram while simultaneously sending these to the strict zero map in the model category. This involves certain simplifications of the general definition, as illustrated in the motivating examples of (long) Toda brackets and Massey products, described in Section 7. Finally, in Section 8 we make a tentative first step towards a possible “algebra of higher operations,” by showing how we can decompose our pointed higher operations into ordinary (long) Toda brackets for a certain class of fully reduced diagrams. In Appendix A we review some basic facts in model categories needed in the paper; Appendix B contains some preliminary remarks on the indeterminacy of the operations. 0.1. Acknowledgements. We wish to thank the referee and editor for their detailed and pertinent comments. The research of the first author was supported by Israel Science Foundation grants 74/11 and 770/16, and the third author by National Science Foundation grant DMS-1207746.

1. The classical Toda Bracket We start with a review of the classical Toda bracket, the primary example of a pointed secondary homotopy operation. In keeping with tradition we give a left justified description, in terms of pushouts, although for technical reasons our general approach will be right justified, in terms of pullbacks. 1.1. Left Justified Toda Brackets. A classical Toda diagram in any pointed model category consists of three composable maps: (1.2)

Y(3)

h

/ Y(2)

g

/ Y(1)

f

/ Y(0)

with each adjacent composite left null-homotopic. We shall assume that all objects in (1.2), and the analogous diagrams throughout the paper, are both fibrant and cofibrant, so we may disregard the distinction between left and right homotopy classes. To define the associated Toda bracket, we first change h into a cofibration (to avoid excessive notation, we do not change the names of h or its target). By Lemma A.11 we can alter g within its homotopy class to a g  to produce a factorization: Y(3) /

(1.3)

h

  / cof(h) 

 ∗ / 0

so g  ◦ h

/ Y(2)

g

g2

  + Y(1)

is the zero map (not just null-homotopic).

24

D. BLANC, M. W. JOHNSON, AND J. M. TURNER

We use i : Y(2) → C Y(2) (an inclusion into a reduced cone) to extend (1.3) to the solid diagram: Y(3) /

h

 ∗ / g

/ Y(2) /

/ C Y(2) U R O

  / / cof(h)  g2

  Y(1) /

i

L H C   / Σ Y(3) / ∗ =φ8 0   3 4 j 0 6 9   /  / cof(g2 ) Mg * A J ( M Iψφ Q κ O % T WT YV X\ ,. #  f Y(0)

where all squares (and thus all rectangles) are pushouts, with cofibrations as indicated. In particular, Σ Y(3) is a model for the reduced suspension of Y(3), Mg is a mapping cone on g  , and φ is a nullhomotopy for f ◦ g  . Note that any choice of such a nullhomotopy φ induces maps ψφ : Σ Y(3) → Y(0) and κ : Mg → Y(0), with κ◦j = ψφ . Suppose that for some choice of φ, the map ψφ is null-homotopic, so κ ◦ j = ψφ ∼ 0. Then by Lemma A.11, we could alter κ within its homotopy class to κ such that κ ◦ j = 0, whence the pushout property for the lower right square would induce the dotted map cof(g2 ) → Y(0). As a consequence, choosing f  = κ ◦ i ∼ κ ◦ i = f provides a replacement for f in the same homotopy class satisfying f  ◦ g  = κ ◦ i ◦ g  = 0, rather than only agreeing up to homotopy. 1.4. Definition. Given (1.2), the subset of the homotopy classes of maps [Σ Y(3), Y(0)] consisting of all classes ψφ (for all choices of φ and g2 as above) forms the Toda bracket f, g, h . Each such ψφ is called a value of f, g, h , and we say that the Toda bracket vanishes (at ψφ : Σ Y(3) → Y(0) as above) if ψφ ∼ ∗ – that is, if f, g, h includes the null map. 1.5. Remark. By what we have shown, f, g, h vanishes if and only if we can vary the spaces Y(0), . . . , Y(3) and the maps f, g, h within their homotopy classes so as to make the adjacent composites in (1.2) (strictly) zero, rather than just null-homotopic. In fact, by considering the cofiber sequence Y(3) → Y(2) → cof(h) → Σ Y(3) one can show that f, g, h is a double coset in the group [Σ Y(3), Y(0)]: the choices for homotopy classes of a nullhomotopy for any fixed pointed map ϕ : A → B are in one-to-one correspondence with classes [ΣA, B] (see [Sp1, §1]), and thus the contribution of the choices for φ and g2 respectively to the value of f, g, h are given by (Σ h)# [Σ Y(2), Y(0)] and f# [Σ Y(3), Y(1)], respectively. The two subgroups (1.6)

(Σ h)# [Σ Y(2), Y(0)]

and

f# [Σ Y(3), Y(1)],

of [Σ Y(3), Y(0)] are referred to as the indeterminacy of f, g, h ; when Y(3) is a homotopy cogroup object or Y(1) is a homotopy group object, the sum of (1.6) is a subgroup of the abelian group [Σ Y(3), Y(0)].

A CONSTRUCTIVE APPROACH TO HIGHER HOMOTOPY OPERATIONS

25

In any case, vanishing means precisely that the (well-defined) class of f, g, h in the double quotient [(Σ h)# Σ Y(2), Y(0)]\[ΣY(3), Y(0)]/f# [Σ Y(3), Y(1)] is the trivial element in the quotient set. 1.7. Remark. The ‘right justified’ definition of our ordinary Toda bracket is given in Step (c) of Section 7.A below. This will depend on a specific initial choice of maps f and g with f ◦ g = ∗ (rather than f ◦ g ∼ ∗), and will be denoted by f, g, h , so  f, g, h = f, g, h f ◦g=∗

where the union is indexed over those pairs with f and g in the specified homotopy classes. The reader is advised to refer to that section for examples of all constructions in Sections 3-4 below, since the example of our long Toda bracket f, g, h, k in Section 7 was the template for our more general setup. 2. Graded Reedy Matching Spaces Our goal is now to extend the notions recalled in Section 1 – of Toda diagrams, and Toda brackets as obstructions to their (pointed) realization – to more general diagrams Y : J → E, where E is some complete category (eventually, a pointed model category). 2.A. Reedy indexing categories Since our approach will be inductive, we need to be able to filter our indexing category J , for which purpose we need the following notions. Recall that a category is said to be locally finite if each Hom-set is finite. 2.1. Definition. We define a weak lattice to be a locally finite Reedy indexing category J (see [Hi, 15.1]), equipped with a degree function deg : Obj J → N, written |x| = deg(x), such that: • J is connected, • there are only finitely many objects in each degree, • all non-identity morphisms strictly decrease degree, and • every object maps to (at least) one of degree zero. 2.2. Remark. Note that a weak lattice J has no directed loops or non-trivial endomorphisms, and x ∈ Obj J has only Idx mapping out of it if and only if |x| = 0. Moreover, each object is the source of only finitely many morphisms, although there may be elements of arbitrarily large degree. 2.3. Notation. For a weak lattice J as above: (a) We denote by Jk the full subcategory of J consisting of the objects of degree ≤ k, with Ik : Jk → J the inclusion. (b) For any x ∈ Obj J in a positive degree, J x will denote the full subcategory of J whose objects are those t ∈ J with J (x, t) nonempty. Thus x ∈ J x and J x ∩ J0 = ∅ (by §2.1).

26

D. BLANC, M. W. JOHNSON, AND J. M. TURNER

(c) We denote by Jkx the full subcategory of J x containing x and all objects (under x) of degree at most k, with Ikx : Jkx → J x the inclusion. We implicitly assume that |x| > k when we use this notation. Similarly, ∂Jkx is the full subcategory of Jkx containing all objects other than x. x → E we have maps (d) Given |x| ≥ k > 0 and a functor Y : Jk−1   x x σk−1 : Y(x) → Y(t) and σ 1) have the correct homotopy type. However, by assumption all such objects Y (s) are fibrant, so we can use the left lifting property for / Y (s) 7 p p α  p ∼ pp  pp  /∗ Y (x) Y(x) 

α

to ensure that α and α  have the same homotopy class. In the inner induction on k, we build up the diagram under the fixed x ∈ Obj J x by extending Yk−1 to objects in degree k, using: x x : Jk−1 →E 3.6. Lemma. Assume |x| > k. Given Yk−1 s g ∈ J (x, s) induces a map ρ(g) : Y(x) → Mk−1 .

and |s| = k, any

A CONSTRUCTIVE APPROACH TO HIGHER HOMOTOPY OPERATIONS

35

x x Proof. Given g, the diagram Yk−1 induces a cone on (s ↓ Jk−1 ), sending x f : s → v to the value of Yk−1 at the target of f g. Moreover, given a morphism

s ~~  ~ ~ f ~~ ~~ ~  /u v f

h

in (s ↓

x Jk−1 ),

precomposition with g yields x ~~  ~ f g ~~ ~~~  /u v fg

h

x x which commutes in J – that is, a morphism in Jk−1 . Applying Yk−1 yields a commutative diagram in E, showing that we have a cone, and thus a map ρ(g) to the limit.  x 3.7. Corollary. Combining all maps ρ(g) of Lemma 3.6, a functor Yk−1 :  s x Mk−1 . Jk−1 → E induces a natural map ρk−1 : Y(x) → J (x,s) |s|=k

3.8. Definition. A pullback grid is a commutative diagram tiled by squares where each square, hence each rectangle in the diagram, is a pullback. and mxk−1

Next, we embed the maps ρk−1 apply Lemma 2.8:

in a pullback grid, in order to

x x 3.9. Lemma. Assuming |x| > n ≥ k ≥ 2, any functor Yk−1 : Jk−1 → E induces a pullback grid defined by the lower horizontal and right vertical maps, with the natural (dashed) maps into the pullbacks:

Y(x) Y B B

W B

βk−1

(3.10)

mx k−1

B

U

R

ηk−1

B

B! Nxk−1

ρk−1

O

LK

qk−1



I# / Qx k−1

/

v



' J (x,s) |s|=k



u

#   Mxk−1 

forget

/

 J (x,t) |t| 0.4 (4) We could mix these procedures, getting an obstruction theory for associative ring spectra based on mod-p homology that lives in Andr´e–Quillen cohomology groups—for algebras in the category of A∗ -comodules. (5) We should mention, at least in passing, the possibility of using a nonconstant operad O• —for example, O• could be a simplicial resolution of the commutative operad. In the commutative case this tends to lead to an obstruction theory closely related to Robinson’s obstruction theory, whose obstruction groups are Γ-cohomology groups [Rob03]. These obstruction groups have been examined in detail for BP in [Ric06], and do not take the Dyer–Lashof operations into account. Part of the goal of this paper is to develop an obstruction theory which does. Remark 3.1. The reader might wonder why we even bother to mention cohomology groups other than those containing obstructions. It is worth pointing out 3 Proving that this is homotopically adapted is now more work, and fails if we try to replace “associative” with “commutative” because the homotopy groups of the free commutative ring spectrum on X are a more confusing functor of π∗ X. 4 Again, there is nothing special about S here, and we could apply this to create an obstruction theory for algebras over a commutative ring spectrum R or differential graded algebras over a commutative ring.

186

TYLER LAWSON

that these groups do more: the groups Exts (E∗ X, Ωt E∗ Y ) serve as a tool for calculating the homotopy groups πt−s Map(X, Y ) for spaces of maps between two realizations [Bou03]. In the above discussion, these specialize to things such as the universal coefficient spectral sequence and the Adams spectral sequence. 4. Homology-based obstructions to commutativity In this section we will discuss a specialization of the Goerss–Hopkins obstruction theory developed by Senger [Sen], whose full writeup is still forthcoming. Just as Serre’s method is improved to Adams’ by switching from a technique that proceeds one homotopy group at a time to one that uses all the cohomology information simultaneously, the Postnikov-based obstruction theory is sometimes improved by the Goerss–Hopkins method that can use both the Dyer–Lashof and Steenrod information simultaneously. To set up this obstruction theory, we need a simplicial operad O• (which we choose to be a constant E∞ -operad) and a homology theory (which we choose to be mod-p homology H∗ ). A simplicial O• -algebra is then a simplicial E∞ ring spectrum, and our “basic cells” are free algebras on finite spectra. Mod-p homology satisfies the Adams–Atiyah condition, and the fact that this operad is homotopically adapted amounts to the following theorems. Theorem 4.1 ([BMMS86, §III.1]). For an E∞ ring spectrum R, the mod-p homology H∗ R has the following structure. (1) It is a comodule over the mod-p dual Steenrod algebra. (2) It is a graded-commutative ring. (3) It has Dyer–Lashof operations that satisfy the Cartan formula, Adem relations, and instability relations. (4) It satisfies the Nishida relations. Following the literature, we will refer to such algebras as AR-algebras.5 Theorem 4.2 ([BMMS86, §IX.2]). The following results about AR-algebras hold: (1) The forgetful functor from AR-algebras to graded comodules has a left adjoint Q. (2) If a comodule M has a basis over Fp of elements ei in degrees ni , then Q(M ) is a free graded-commutative algebra on the elements P (ei ) such that P is an admissible monomial in the Dyer–Lashof algebra of excess at least ni . (3) If we write P(X) for the free E∞ ring spectrum on X, then the homology H∗ P(X) is a free AR-algebra: the natural map H∗ X → H∗ P(X) induces a natural isomorphism Q(H∗ X) → H∗ P(X). 5 Because the origin of this subject is in studying the homology of infinite loop spaces rather than the homology of E∞ ring spectra, the definitions of AR-algebras in the literature often involve connectivity assumptions and only discuss Dyer–Lashof operations of nonnegative degree.

CALCULATING OBSTRUCTION GROUPS FOR E∞ RING SPECTRA

187

As a result, the mod-p homology of simplicial E∞ ring spectra takes place in the category of simplicial AR-algebras. The Ext-groups of an AR-algebra B∗ here have coefficients in an AR-B∗ -module: a B∗ -module in A∗ -comodules, with compatible Dyer–Lashof operations Qs such that Qs (x) = 0 for s ≤ |x|. Theorem 4.3 ([Sen]). Given B∗ , an AR-algebra, there are Goerss–Hopkins obstruction groups DersAR (B∗ , Ωt B∗ ) calculated in the category of simplicial AR-algebras. The groups with t − s = −2 contain an iterative sequence of obstructions to realizing B∗ by an E∞ ring spectrum R such that H∗ R ∼ = B∗ , and the groups with t − s = −1 contain obstructions to uniqueness. From this point forward, it will be our goal to give methods to calculate the nonabelian Ext-groups DersAR in specific cases and to interpret elements in them as concrete obstructions. 5. Tools for calculation In this section, we will begin discussing how [Sen] reduces the Goerss–Hopkins obstruction theory for the Brown–Peterson spectrum BP and its truncated versions BP n to more straightforward calculations.6 We need to calculate the obstruction groups DersAR (H∗ BP, Ωt H∗ BP ).7 We begin by recalling the structure of H∗ BP as a comodule over the dual Steenrod algebra. The dual Steenrod algebra A∗ has quotient Hopf algebras, given by exterior algebras E(n)∗ = Λ[τ0 , τ1 , . . . , τn ] i with |τi | = 2p − 1. (At the prime 2, τi is the image of ξi .) When n = ∞, we get a Hopf algebra E∗ . The homology H∗ BP n can be identified with a coextended comodule: H∗ BP n ∼ = A∗ E(n)∗ Fp . This coextension functor is right adjoint to the forgetful functor from A∗ -comodules to E(n)∗ -comodules. To proceed, we need to know that this adjunction is compatible with Dyer–Lashof operations. Proposition 5.1 ([Bak15, 7.3]). Let p = 2, and let Mr be the Milnor primitive of degree 2r − 1 dual to ξr in the Milnor basis of the dual Steenrod algebra. Then the Nishida relations imply  r r k (5.1) Mr Qs = (s + 1)Qs−2 +1 + Qs−2 +2 Mk . 0≤k 0, this localization defines a version Kalg (P )∗ (−, Zp ) → H ∗ (−, γP−1 THH∗ (OP , Qp )) of the classical Chern-Dold character. There is a large literature (e.g. Segal [28], Snaith [31], Boyer, Lawson, Lima-Filho, Mann and Michelson [8], Totaro [34, 35] . . . ) on related integrality questions. §III Generalized cyclotomic fields Any L ⊃ Qp as in §1.3.1 admits a maximal totally ramified Abelian extension L∞ ⊃ L, with an Artin reciprocity homomorphism [29] Gal(Qp /L)

 × Gal(L∞ /L) /Z

/ Gal(L∞ /L)

κL

/ OL × ,

∼ where Z = Gal(Lnr /L) ∼ = Gal(kL /kL ), which classifies unramified extensions, has ˆ of L∞ in been supressed; they will play no part in this paper. The completion L∞ Cp is perfectoid; see [22, §1.4.17] or [37, Ex. 2.0.4, Ex. 2.1.1] for its tilt, along with much more information3 . ∗ ˆ ) (Bμ, Zp ) defined above We argue below that the graded formal groups THH(OL∞ have a natural interpretation in terms of the p-adic Fourier theory of Schneider and Teitelbaum, as a rigid analytic version [23, §5 intro.]

LTrigid L  Spec Cp

εL

/ LTL  / Spec OL

of the Lubin-Tate formal groups LTL used to construct these extensions, mapped by an analog of the exponential map of classical Lie theory (defined only in a neighborhood of the origin). 3.1.1 The construction of LTL is elegant and in some sense quite elementary, but it depends (up to a canonical isomorphism) on some choices, i.e. of a uniformizing element πL as in §1.3.1, as well as an element [πL ](T ) ∈ OL [[T ]] equal to πL T modulo terms of higher order, and congruent modulo mL to T q . In the following we will assume that LTL is special in the sense of Lang, i.e. that [πL ](T ) = πL T + T q ; 3 To a topologist it is tempting to call these ‘chromatic’ fields. There are interesting analogies with the Alexander cover of a link complement . . .

COMPLEX ORIENTATIONS FOR THH OF SOME PERFECTOID FIELDS

229

this implies that T → ωT , for ω ∈ W (kL )× ⊂ OL × , is an automorphism of LTL , which simplifies issues of grading in homotopy theory. The resulting formal group X, Y → FL (X, Y ) := X +L Y ∈ OL [[X, Y ]] is in fact a formal OL -module, endowed with an isomorphism a → [a]L : OL → EndOCp (LTL ) . The group HomOL ,c (OL [[T ]], OCp ) =: LTL (mCp ) = (mCp , +L ) of points of LTL (defined by continuous homomorphisms) is isomorphic (modulo Q-vector spaces) to (Qp /Zp )n , and its Tate module TL := Hom(Qp /Zp , LTL (mCp )) is free of rank one over End(LTL ) ∼ = OL . Adjoining the torsion points of LTL (mCp ) to L defines the extension L∞ ⊃ L; the Galois group Gal(Qp /L) acts on these torsion points, defining a reciprocity map Gal(Qp /L) → Gal(L∞ /L) ∼ = AutO (TL ) ∼ = OL × . L

If L0 ⊂ L1 is Galois, then the diagram [33] 1

/ Gal(Qp /L0 )

/ Gal(Qp /L1 ) 

κL1

(L× 1 )ˆ

N01



κL0

/ (L× )ˆ 0

/ Gal(L1 /L0 )

/1

 / Gal(L1 /L0 )ab

(with N01 = NLL01 the norm, and profinite completion denoted by a caret) commutes. In particular κQp (σ) = NQLp (κL (σ)) if σ ∈ Gal(Qp /L). Reducing modulo mL defines a formal group law on kL [[T ]] of height n, and thus × × an embedding of OL in the automorphism group OD of the reduction as a kind of maximal torus. 3.1.2 A formal group law (e.g. LTL ) over a torsion-free ring (e.g. unique logarithm and exponential, e.g.

OL ) has a

logL (T ), expL (T ) ∈ L[[T ]] such that X +L Y = expL (logL (X) + logL (Y )) ∈ OL [[X, Y ]] . The formal multiplicative group of §2.1.2 is one classical example, and Honda’s logarithm [13]  n logπ (T ) := π −n T q ∈ L[[T ]] n≥0

is another. Lemma 2.1.3 generalizes to special Lubin-Tate groups as follows: Definition Let π0 ∈ LTL (mCp ) be a primitive [π]-torsion point, i.e. a generator of the cyclic group of points satisfying π0q−1 + π = 0 , × ∼  := L(π0 ); then Gal(L/L)  and let L ⊂ W (kL )× (using Teichm¨ uller represen= kL tatives).

230

JACK MORAVA

Proposition The formal group law X +L Y := π0−1 (π0 X +L π0 Y ) ∈ OL [[X, Y ]] is of additive type, ie with [p]L (T ) ≡ 0 modulo π0 . Proof Evidently [π]L (T ) ∼ = π(T − T q ) ∼ = 0 modulo π0 . But [π]L satisfies an Eisenstein equation EL ([π]L ) = [π]eL +L · · · +L [u]L ◦ [p]L = 0 (with coefficients from W (kL ), and u a unit) in End(LTL ), so, similarly, [π]eL +L · · · +L [u]L ◦ [p]L = 0 in the endomorphisms of FL . But [π]L ≡ 0 mod π0 , so [p]L ≡ 0 mod π0 , as well.  Example Honda’s logarithm is p-typical, so its renormalization  n n π0−1 logπ (π0 T ) = ±π0q −1 π −n T q n≥0

has coefficients in OL , with ordp (π0q

n

−1 −n

π

) = e−1 (

qn − 1 − n) = e−1 (1 + · · · + q n−1 − n) ≥ 0 , q−1

which goes to ∞ as n does, making it a rigid analytic function. Note that the completed Hopf OL -algebra defined by FL is the pushforward, under the homomorphism defined by T → π0 T : OL [[T ]] → OL [[T ]] , of that defined by FL ; but (because this map does not preserve the coordinate) it is not a morphism of formal group laws. 3.2.1 A Lubin-Tate group has an associated p-divisible group with Cartier dual ˆ m ); early work of Katz [14, §3] identifies its Tate module Hom(LTL , G TL∨ := {β(T ) ∈ (1 + T OCp [[T ]])× | β(X +L Y ) = β(X) · β(Y )} as free of rank one over OL . A homotopy theorist will recognize this as a specialization to Lubin-Tate groups of Ravenel and Wilson’s almost simultaneous description [24] of the Hopf algebra MU∗ (CP ∞ ) representing the Cartier dual of the universal p-divisible group: the canonical inclusions CP i ⊂ CP ∞ define bordism classes bi with  b(T ) = 1 + bi T i i>0

satisfying the relation b(X +MU Y ) = b(X) · b(Y ) under the Pontryagin product. Classical Fourier analysis identifies the dual V ∗ := HomR (V, R) of a finite-dimensional vector space V with its character group V X := Homc (V, T) by V ∗ & ξ → [x → exp(iξ(x))] ∈ V χ .

COMPLEX ORIENTATIONS FOR THH OF SOME PERFECTOID FIELDS

231

In 2001 Schneider and Teitelbaum [26, §2] defined a p-adic analog of the Pontryagin dual of a free OL -module M as the rigid analytic group M X of locally analytic characters θ : M → (1 + mCp )× , and showed that the map X ∼ LTL (mCp ) & α → [β → [a → β([a]L (α))]] ∈ HomOL (TL∨ , OL ) = (TL∨ )X

is an isomorphism of Gal(Qp /L)-modules. The inverse of this equivalence defines the morphism := (TL∨ )X → LTL εL : LTrigid L of group-valued functors cited above, represented by a homomorphism of (completed) Hopf OL -algebras from OL [[T ]] to locally analytic homomorphisms [30, §6.1] from TL∨ to (1 + mCp )× , sending T to εL (T ) = expGˆ m (Ω(L) logL (T )) ∈ Lan (OL , Cp ) : where Ω(L) ∈ oˆL∞ , with ordp (Ω(L)) = (p − 1)−1 − e−1 (q − 1)−1 , is a ‘period’ of the formal group LTL , with the remarkable property that κQ × Ω(L)σ−1 = { p }(σ) ∈ OL κL for σ ∈ Gal(L∞ /L) [3,26, lemma 3.4, 27, §6.2.3 Prop 6.4]. This can be reformulated as the assertion that κQ ˆ × ) ), Hc1 (Gal(L∞ /L), OL × ) & { p } → 0 ∈ Hc1 (Gal(L∞ /L), (L∞ κL or as the 3.2.2 Proposition The diagram (in which the vertical arrows multiply T by the indicated element) εL ˆ / L∞ L[[T ]] [[T ]] p0

π0

  ]] L[[T

ε0L

 ˆ / L∞ [[T ]]

commutes, where ε0L (T ) := expG˜ m (Ω0 (L) logL (T )) = Ω0 (L)T + · · · ∈ OL∞ ˆ [[T ]] , and Ω0 (L) := p0 π0−1 Ω(L) is a unit. Moreover, the map T → ε0L (T ) is equivariant, in the sense that 0 0 [κQp (σ)]−1  = σ(εL ) , ˜ ◦ εL ◦ [κL (σ)]L G m

with respect to the action of σ ∈ Gal(L∞ /L) on the coefficients of ε0L (T ). Proof We have logL ([κ]L (σ)]L (T )) = κL (σ) logL (T ) and Ω(L)κL (σ) κQp (σ)σ(Ω(L)); while expG˜ m (κQp (σ) · · · ) = [κQp (σ)]G˜ m (· · · ). 

=

3.3.1 In 1982 Fontaine [9, Thm. 1] defined, for a Lubin-Tate group of L, a homomorphism ξL : Qp ⊗OL TL → Ω1OL (OQp )

232

JACK MORAVA

of modules over the twisted group ring4 OQp  Gal(Qp /L) : if h = {hn } ∈ TL with [π n ]L (hn ) = 0, and Qp & α = αn π n with αn ∈ OQp integral, then ξL (α ⊗ h) := αn · h∗n (d logL (T )) ∈ Ω1OL (OQp ) is well-defined; where hn ∈ LTL (mQp ), and d logL (T ) = logL (T ) · dT . He then proves the exactness of the sequence / π −1 O

0

Qp

0

/ Qp ⊗OL TL

⊗OL TL

and taking Tate modules defines an isomorphism T Ω1 (O ) ∼ = p−1 OC ⊗O OQp

Qp

0

p

/ Ω1 (O ) OL Qp

ξL

Qp

/0 ,

TQp

of Gal(Qp /Qp )-modules. This argument generalizes, without significant change, to imply the existence of an exact sequence / π −1 OL∞ ⊗O TL L 0

0

/ L∞ ⊗O TL L

ξL

/ Ω1 (OL∞ ) OL

/0

of Gal(L∞ /L)-modules, and hence an isomorphism Ω1OL (OL∞ ) ∼ = (L∞ /π0−1 OL∞ ) ⊗OL TL .

3.3.2 Now a sequence Qp ⊂ L ⊂ L ⊂ L ⊂ Cp of extensions implies [9, §2.4 lemma 2] a monomorphism 0 → Ω1OL  (OL ) → Ω1OL  (OL  ) , and since OL∞ is flat over OL , the Jacobi-Zariski exact sequence 0 → Ω1OQp (OL ) ⊗OL OL∞ → Ω1OQp (OL∞ ) → Ω1OL (OL∞ ) → 0 implies that

ˆ ⊗O TL T Ω1OQp (OL∞ ) ∼ = (π0 DL )−1 OL∞ L

−1 as Gal(L∞ /Qp )-modules; where Ω1OQp (OL ) ∼ /OL (the Dedekind = DL = OL /DL OL ∼ different being the inverse of the fractional ideal HomZp (OL , Zp )) defined by the trace from OL to OQp .

Similarly, since Qp is flat over Q∞ p , the monomorphism 0 → Ω1OQp (OQ∞ ) ⊗OQ∞ OQp → Ω1OQp (OQp ) p p implies a commutative diagram / Ω1 (O ) OQp Qp

Ω1OQp (OQ∞ ) ⊗OQ∞ OQp p p 

∼ =

Qp /p−1 0 OQp

∼ =



/ Ω1

(OQp ) OQ∞ p

/0

∼ =

/ Qp /p−1 O ; 0 Qp

 ∞ /L) as elements, and multiplication finite sums aσ · σ, aσ ∈ OL∞ ˆ , σ ∈ Gal(L (a · σ)(b · τ ) = aσ(b) · στ 4 with

COMPLEX ORIENTATIONS FOR THH OF SOME PERFECTOID FIELDS

233

but then Ω1OQ∞ (OQp ) = 0, and hence (since p

0 → Ω1OQ∞ (OL∞ ) → Ω1OQ∞ (OQp ) = 0 p

p

is injective), that Ω1OQ∞ (OL∞ ) = 0. p

Corollary The exact sequence 0 → Ω1OQp (OQ∞ ) ⊗OQ∞ OL∞ → Ω1OQp (OL∞ ) → Ω1OQ∞ (OL∞ ) = 0 p p p

implies an isomorphism φ : p−1 ˆ ⊗O TQp ∼ ˆ ⊗O TL = (π0 DL )−1 OL∞ Qp L 0 OL∞ ∞ of Tate modules over OL∞ /Qp ) . ˆ  Gal(L

Proposition Let tQp , tL generate the Tate modules TQp , TL , and let σ ∈ Gal(L∞ /L); then ˆ t−1 L φ(tQp ) := Ω∂ (L) ∈ OL∞ also satisfies Ω∂ (L)σ−1 = {

κQp }(σ) , κL

and hence DL Ω∂ (L) = u · Ω(L) for some unit u = u(L : Qp ) ∈ OL × . With tQp as in §1.3, we can choose tL so Ω∂L ≡ 1 mod mL∞ ˆ (and in particular so (Ω∂L )k → 1 as k → ∞). The Galois action on THH of the Teichm¨ uller units in W (kL ) is then consistent with the topological grading. §IV Applications and speculations Writing ε∂L for the variant of ε0L defined by replacing Ω0 (L) by Ω∂0 (L) defines a composition 0 Spf THH(OL∞ ˆ ) (Bμ, Zp )

DL

˜ m ×O OL∞ /G ˆ  L

ε∂ L

/ F L

 Spec OL∞ ˆ

π0

/ LTL  / Spec OL

of affine group schemes, with the top right morphism as in §3.1.2 and the top left morphism defined (following §2.1.4) by T → DL T . By §3.2.2, the morphism across × the top takes the natural action of Gal(L∞ /L) on the left to the action of OL by formal group automorphisms on the right, compatible with Artin reciprocity, and it seems natural to conjecture that when L/Qp is Galois, this extends to equivariance with respect to an action of Gal(L∞ /Qp ); see Appendix III of [36]. The morphism χL from the Lazard ring to OL which classifies LTL lifts to a graded Hirzebruch genus MU∗ & [M ] → χL [M ] · udimC M ∈ OL [u] ,

234

JACK MORAVA

defining a commutative diagram / OL [u] := k∗ (L) MU∗ SS SSS SSSκL∞ SSSˆS u→π0 DL γL∞ ˆ SSS S)  ∼ THH∗ (OL∞ ˆ , Zp ) = OL∞ ˆ [γL∞ ˆ ] χL

of ring homomorphisms. The appendix below summarizes a construction for weakly commutative complex-oriented cohomology theories with Spf k(L)0 (CP ∞ ) ∼ = LTL . Hirzebruch’s work from the 60s, interpreting multiplicative natural transformations [MU∗ → E ∗ ] ∈ E 0 (MU) ∼ = E 0 (B U) of cohomology theories in terms of the Thom isomorphism and symmetric functions, sending an orientation κ ∈ E 0 (BT) to the (grouplike, i.e. Δκ = κ ⊗ κ) limit κ (as k 0 k → ∞ of ϕ−1 E (⊗ κ) ∈ E (M U(k))), defines a lift of this diagram to a diagram of multiplicative natural transformations between Z2 -graded cohomology theories. Acknowledgements and thanks This paper has roots in memorable conversations with B¨ okstedt and Waldhausen, in Bielefeld during the Chernobyl weekend: they had just invented topological Hochschild homology, and although they could see glimmers of connection with chromatic homotopy theory [7, 20], it was Lars Hesselholt’s talk [12] at the 2015 Oxford Clay symposium that really opened the door. I am deeply indebted to him for his generous help and patient willingness since then to endure iterated attempts to misunderstand his work. I hope I have not continued to garble it, and I want to thank an extremely perceptive and insightful referee for help in clarifying a very rough early version of this paper. I also owe thanks to Matthias Strauch for calling my attention to the relevance of p-adic Fourier theory for this project, to Peter Schneider for his course notes, to Jacob Lurie, Akhil Mathew and Chuck Weibel for conversations about the descent properties of THH, to Andy Baker for arithmetic and infinite loop-space counsel, and to Peter Scholze for sharing his insights and results, especially as discussed in §1.3.2. I am indebted as well to the Mittag-Leffler Institute for the opportunity for more conversations with Hesselholt, and to Mona Merling, Apurv Nakade, and Xiyuan Wang at JHU for their helpful attention. Finally I want to thank the mathematics department at UIUC for organizing this conference, and beyond that, to acknowledge deep lifetime debts to both Paul Goerss and Matt Ando, two subtle, quiet enormous figures who loom, almost invisibly, over so much of the work that has made the research world we live in today. Appendix: Integral lifts of K(n) parametrized by local fields For any n ∈ N and prime p (or, alternately, for every field with q = pn elements), there is a functor K(n) : (Spaces) → (Fq − modules)

COMPLEX ORIENTATIONS FOR THH OF SOME PERFECTOID FIELDS

235

(roughly, the residue field at a prime of the sphere spectrum) with K(n)∗ (BT) = Fq [u±1 ][[T ]] a graded complete Hopf algebra or formal group with addition +K(n) , with |u| = +2, |T | = −2, such that the p-fold multiplication map is represented by [p]K(n) (T ) = uq−1 T q . Araki’s generators for BP∗ (pt) = Z(p) [. . . , vi , . . . ] satisfy  i [p]BP (T ) = vi T p = pT +BP v1 T p +BP . . . BP

(i ≥ 0, v0 = p), and there is a Baas-Sullivan quotient ([18, §3.4]: the totalization of a suitable Koszul complex, or an iterated homotopy cofiber) with MU∗ (pt) → BP∗ (pt) → Zp [u] , such that vi → 0, i = 0, n and vn → uq−1 , classifying a graded group law +k(n) with n [p]k(n) (T ) = pT +k(n) uq−1 T p , i.e. +k(n) ≡ +K(n) mod p. Hazewinkel’s functional equation [22] implies that T0 +k(n) T1 = u−1 expk(n) (logk(n) (uT0 ) + logk(n) (uT1 )) ∈ Zp [u][[T0 , T1 ]] with logk(n) (uT ) := uT +

 

(1 − pq

i

−1 −1

)

· p−k (uT )q ∈ Qp [[uT ]] . k

k>0 1≤i≤k

Tensoring this Baas-Sullivan theory with W (Fq ) defines a cohomology theory k(L0 ) (where L0 := W (Fq ) ⊗ Q is the unique unramified extension of degree n of Qp ), such that k(L0 )∗ (pt) = W (Fq )[u] → Fq [u] → Fq [u±1 ] = K(n)∗ (pt) defines a nice integral lift to a connective version of K(n). More generally, for any L ⊂ Qp with n = [L : Qp ], there is a connective spectrum k(L) with k(L)∗ (pt) = OL [u], and such that k(L)∗ (BT) is a Lubin-Tate formal group for L; e.g. if n = 1, k(Qp ) is the p-adic completion of connective classical topological K-theory (associated to the multiplicative formal group). Lubin-Tate groups of local number fields parametrize good integral lifts of K(n). × In particular, the group OL of units acts as stable multiplicative automorphisms of × ∗ ∞ k(L) (CP ), e.g. with α ∈ OL sending u to α · u (generalizing the action of Zp × by p-adic Adams operations on classical (p-completed) topological K-theory). It seems natural to think of these lifts as indexed by maximal toruses in the unit group Dn× of a division algebra with center Qp and Brauer-Hasse invariant 1/n. Recent work [18] of Hopkins and Lurie, using the modern theory of Thom spectra, has changed the geography of this subject: in particular, it raises the question of possible Azumaya multiplications on such lifts, which might support lifts of the Gal(L∞ /L)-action discussed above, to an action compatible with some such multiplicative structure.

236

JACK MORAVA

References [1]

[2] [3] [4] [5] [6] [7] [8]

[9]

[10] [11]

[12] [13] [14]

[15] [16]

[17] [18] [19] [20]

[21] [22] [23] [24] [25]

[26]

M. Ando and J. Morava, A renormalized Riemann-Roch formula and the Thom isomorphism for the free loop space, Topology, geometry, and algebra: interactions and new directions (Stanford, CA, 1999), Contemp. Math., vol. 279, Amer. Math. Soc., Providence, RI, 2001, pp. 11–36, DOI 10.1090/conm/279/04552. MR1850739 M. F. Atiyah and D. O. Tall, Group representations, λ-rings and the J-homomorphism, Topology 8 (1969), 253–297, DOI 10.1016/0040-9383(69)90015-9. MR0244387 A. Baker, p-adic continuous functions on rings of integers and a theorem of K. Mahler, J. London Math. Soc. (2) 33 (1986), no. 3, 414–420, DOI 10.1112/jlms/s2-33.3.414. MR850957 B Bhatt, Completions and derived de Rham cohomology, available at http://arxiv.org/ abs/1207.6193 B Bhatt, M Morrow, P Scholze, Integral p-adic Hodge theory, available at arXiv:1602.03148 B Bhatt, M Morrow, P Scholze, Topological Hochschild homology and integral p-adic Hodge theory, available at arXiv:1802.03261 M B¨ okstedt, Topological Hochschild homology, preprint, Bielefeld (1985) C. P. Boyer, H. B. Lawson Jr., P. Lima-Filho, B. M. Mann, and M.-L. Michelsohn, Algebraic cycles and infinite loop spaces, Invent. Math. 113 (1993), no. 2, 373–388, DOI 10.1007/BF01244311. MR1228130 J.-M. Fontaine, Formes diff´ erentielles et modules de Tate des vari´ et´ es ab´ eliennes sur les corps locaux (French), Invent. Math. 65 (1981/82), no. 3, 379–409, DOI 10.1007/BF01396625. MR643559 L. Hesselholt and I. Madsen, On the K-theory of local fields, Ann. of Math. (2) 158 (2003), no. 1, 1–113, DOI 10.4007/annals.2003.158.1. MR1998478 L. Hesselholt, On the topological cyclic homology of the algebraic closure of a local field, An alpine anthology of homotopy theory, Contemp. Math., vol. 399, Amer. Math. Soc., Providence, RI, 2006, pp. 133–162, DOI 10.1090/conm/399/07517. MR2222509 L Hesselholt, Periodic topological cyclic homology and the Hasse-Weil zeta function, available at https://arxiv.org/abs/1602.01980 T. Honda, On the theory of commutative formal groups, J. Math. Soc. Japan 22 (1970), 213–246, DOI 10.2969/jmsj/02220213. MR0255551 N. M. Katz, Formal groups and p-adic interpolation, Journ´ ees Arithm´ etiques de Caen (Univ. Caen, Caen, 1976), Soc. Math. France, Paris, 1977, pp. pp 55–65. Ast´ erisque No. 41–42. MR0441928 S. Lang, Cyclotomic fields. II, Graduate Texts in Mathematics, vol. 69, Springer-Verlag, New York-Berlin, 1980. MR566952 A. Lindenstrauss and I. Madsen, Topological Hochschild homology of number rings, Trans. Amer. Math. Soc. 352 (2000), no. 5, 2179–2204, DOI 10.1090/S0002-9947-00-02611-8. MR1707702 J Lurie, M Hopkins, Harvard Thursday seminar 2015-6, notes by A Mathew J Lurie, M Hopkins, On Brauer groups of Lubin-Tate spectra I, available at http://www. math.harvard.edu/~lurie/ JP May, Topological Hochschild and cyclic homology and algebraic K-theory, available at http://www.math.uchicago.edu/~may/TALKS/THHTC.pdf J. Morava, Browder-Fr¨ ohlich symbols, Algebraic topology (Arcata, CA, 1986), Lecture Notes in Math., vol. 1370, Springer, Berlin, 1989, pp. 335–345, DOI 10.1007/BFb0085238. MR1000387 J Morava, Local fields and extraordinary K-theory, available at arxiv:1207.4011 T Nikolaus, P Scholze, On topological cyclic homology, available at arXiv:1707.01799 M. Rapoport and Th. Zink, Period spaces for p-divisible groups, Annals of Mathematics Studies, vol. 141, Princeton University Press, Princeton, NJ, 1996. MR1393439 D. C. Ravenel and W. S. Wilson, The Hopf ring for complex cobordism, J. Pure Appl. Algebra 9 (1976/77), no. 3, 241–280, DOI 10.1016/0022-4049(77)90070-6. MR0448337 P Schneider, Galois representations and (φ, Γ)-modules, Cambridge Studies in Advanced Math (2017), available at http://www.math.uni-muenster.de/u/schneider/publ/ lectnotes/index.html P. Schneider and J. Teitelbaum, p-adic Fourier theory, Doc. Math. 6 (2001), 447–481. MR1871671

COMPLEX ORIENTATIONS FOR THH OF SOME PERFECTOID FIELDS

237

[27] P Scholze, p-adic geometry, available at arxiv:1712.03708 [28] G. Segal, The multiplicative group of classical cohomology, Quart. J. Math. Oxford Ser. (2) 26 (1975), no. 103, 289–293, DOI 10.1093/qmath/26.1.289. MR0380770 [29] J.-P. Serre, Local class field theory, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 128–161. MR0220701 [30] E. de Shalit, Mahler bases and elementary p-adic analysis (English, with English and French summaries), J. Th´ eor. Nombres Bordeaux 28 (2016), no. 3, 597–620. MR3610689 [31] V. Snaith, The total Chern and Stiefel-Whitney classes are not infinite loop maps, Illinois J. Math. 21 (1977), no. 2, 300–304. MR0433446 [32] D. Sullivan, Genetics of homotopy theory and the Adams conjecture, Ann. of Math. (2) 100 (1974), 1–79, DOI 10.2307/1970841. MR0442930 [33] J. Tate, Number theoretic background, Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–26. MR546607 [34] B. Totaro, The total Chern class is not a map of multiplicative cohomology theories, Math. Z. 212 (1993), no. 4, 527–532, DOI 10.1007/BF02571672. MR1214042 [35] B. Totaro, Homology operations on a new infinite loop space, Trans. Amer. Math. Soc. 347 (1995), no. 1, 99–110, DOI 10.2307/2154790. MR1273541 [36] A. Weil, Basic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 144, Springer-Verlag New York, Inc., New York, 1967. MR0234930 [37] J Weinstein, Adic spaces, available at swc.math.arizona.edu/aws/2017/ 2017WeinsteinNotes.pdf Department of Mathematics, The Johns Hopkins University, Baltimore, Maryland 21218 Email address: [email protected]

Contemporary Mathematics Volume 729, 2019 https://doi.org/10.1090/conm/729/14698

String bordism and chromatic characteristics Markus Szymik Dedicated to Paul Goerss on the occasion of his 60th birthday Abstract. We introduce characteristics into chromatic homotopy theory. This parallels the prime characteristics in number theory as well as in our earlier work on structured ring spectra and unoriented bordism theory. Here, the K(n)–local Hopkins–Miller classes ζn take the place of the prime numbers. Examples from topological and algebraic K-theory, topological modular forms, and higher bordism spectra motivate and illustrate this concept.

Introduction The classification of manifolds is intimately tied to the homotopy theory of Thom spaces and spectra. If MO denotes the Thom spectrum for the family of orthogonal groups, then its homotopy groups πd MO are given by the groups of bordism classes of d–dimensional closed manifolds. Variants of this correspondence apply to manifolds with extra structure, such as orientations and Spin structures, for instance. Arguably the most relevant of these variants for geometry are ordered into a hierarchy given by the higher connective covers BOk → BO of BO, and their Thom spectra MOk . For small values of k, these describe the unoriented (MO1 = MO), oriented (MO2 = MSO), Spin (MO4 = MSpin), and String bordism groups of manifolds (MO8 = MString). The name ‘string’ in this context appears to be due to Miller (see [34]). The spectra MOk are also interesting as approximations to the sphere spectrum S itself, in a sense that can be made precise [26, Proposition 2.1.1]: There is an equivalence S  limk MOk . The geometric relevance of the sphere spectrum stems, of course, from the fact that it is the Thom spectrum for stably framed manifolds. All the bordism spectra that were just mentioned are canonically commutative ring spectra in the most desirable way, namely E∞ ring spectra [39]. In fact, this concept was more or less invented in order to deal with the very examples of Thom spectra [37]. The multiplicative structure allows us to study them through their genera: multiplicative maps out of commutative bordism ring spectra into spectra which are easier to understand. This has been rather successful for small values of k, and the following diagram indicates the situation.

2010 Mathematics Subject Classification. Primary 55N22, 55P43; Secondary 19L41, 57R90, 58J26. c 2019 American Mathematical Society

239

240

MARKUS SZYMIK

.. .  MString

/ tmf

 MSpin

/ ko

 MSO

/ HZ

 MO

/ HF2

Here, the spectra HF2 and HZ are the Eilenberg–Mac Lane spectra of the indicated rings, and the genera count the number of points mod 2 and with signs, respectively. In the row above, the spectrum ko is the connective real K-theory spectrum that re ceives the topological A–genus (or Atiyah–Bott–Shapiro orientation) (compare [31] and [32]). Finally, the spectrum tmf is the spectrum of topological modular forms that was constructed in order to refine the Witten genus (or σ–orientation) (see [20], [21], [2], [3], and [1]). Characteristics in the sense of the title appear in the approach that is dual to the idea underlying genera. Namely, there are interesting ring spectra that come with maps into these bordism spectra. For instance, since the unoriented bordism ring π∗ MO has characteristic 2, there is a unique (up to homotopy) map S  2 → MO of E∞ ring spectra from the versal E∞ ring spectrum S  2 of characteristic 2. See [48], where E∞ ring spectra of prime characteristics, and their versal examples S  p, have been studied from this point of view. However, the fact that π0 MOk = Z as soon as k  2 makes it evident that ordinary prime characteristics have nothing to say about higher bordism theories. This is where the present writing sets in. See also [7] for a different generalization. In order to gain a better understanding of higher bordism theories, we propose in this paper to replace the ordinary primes p ∈ Z = π0 S by something more elaborate, namely by some classes that only appear after passing to the (Bousfield [14]) localization  S of the sphere spectrum S with respect to any given Morava KS which were first defined by Hopkins and Miller. theory K(n): the classes ζn in π−1  See [25], [16], and our exposition in Section 1. Just as S  2 has been used in [48] to study the unoriented bordism spectrum MO, one aim of the present writing is to show that it is the corresponding versal examples  S  ζn which are likewise relevant to the study of the chromatic localizations of higher bordism spectra. Whenever A is any K(n)–local E∞ ring spectrum with unit uA :  S → A, there is a naturally associated class A ζn (A) : S−1 −→  S −→ A

ζn

u

in π−1 A. Continuing to use the terminology as in [48], we will say that a K(n)– local E∞ ring spectrum A has (chromatic) characteristic ζn if there exists a homotopy ζn (A)  0 (compare Definition 1.12 below). We note that this concept only involves the existence of a homotopy, whereas for structural purposes one will want to work with actual choices of homotopies, i.e. with commutative  S  ζn –algebras. See Section 2.1, and [48] again.

STRING BORDISM AND CHROMATIC CHARACTERISTICS

241

There are families of examples of characteristic ζn spectra for arbitrary n: the Lubin–Tate spectra En (Example 2.2), the Iwasawa extensions Bn of the K(n)–local sphere (Example 2.3), and the versal examples  S  ζn that map to all of these (see Proposition 2.5). Hopkins [22] and Laures [36] have given useful descriptions of the K(1)–local E∞ ring spectra KOK(1) and tmf K(1) at the prime p = 2. The first step in these S in an E∞ manner so as to obtain the versal example cases is to kill the class ζ1 in   S  ζ1 above. The second (and already last) step in either case is to kill another class in the latter. This underlines the importance of an understanding of the versal examples  S  ζn , and since B1 = KOK(1) , it naturally leads one to ask for a similar description of the higher Iwasawa extensions Bn . I hope this will be pursued elsewhere (see Remark 2.7). The K(1)–localizations of many algebraic K-theory spectra are not of characteristic ζ1 , and the behavior of multiplication with ζ1 on the homotopy groups is connected to open number theoretic conjectures (see Remark 3.3). In contrast to that, the work of Laures [35] and his student Reeker [44] shows that the K(1)– localizations of MSpin, and MSU all have characteristic ζ1 . In some genuinely new examples dealt with here, we take the natural next step: The K(2)–localizations of the topological modular forms spectrum tmf, the String bordism spectrum MString, and MU6 have characteristic ζ2 almost everywhere (see Propositions 4.1 and 5.4). The paper is organized into five sections. In Section 1, we briefly review the basic context for chromatic homotopy theory, establish the notation that we are going to use here, and define chromatic characteristics. Section 2 introduces the versal examples and presents the higher Iwasawa extensions. Section 3 contains our discussion of topological and algebraic K-theory spectra. In Section 4, we show how to deal with spectra related to topological modular forms, and bordism spectra are examined in the final Section 5. 1. Characteristics in chromatic homotopy theory In this section we will review some chromatic homotopy theory as far as it is needed for our purposes, and introduce the basic concept of chromatic characteristics (see Section 1.7). The case n = 1 will be mentioned as an accompanying example throughout, but we emphasize that this case is always somewhat atypical, and the general case is the one we are interested in. Also, in the spirit of [30], we have chosen notation that avoids having to say anything special when p = 2. Nevertheless, we do so, if it seems appropriate for the examples at hand, in particular in Section 5 when it comes to bordism theories. We will use the following conventions: All spectra are implicitly K(n)–localized. In particular, the notation X ∧ Y will refer to the K(n)–localization of the usual smash product, and the homology X0 Y is defined as π0 of that. As an exception to these rules, we will write  S for the K(n)–local sphere to emphasize the idea S denotes that it is a completed form of the sphere spectrum S, and Sn = Σn  its (de)suspensions. 1.1. The Lubin–Tate spectra. Let p be a prime number, and n a positive integer. We will denote by En the corresponding Lubin–Tate spectrum. The

242

MARKUS SZYMIK

coefficient ring is isomorphic to π ∗ En ∼ = W(Fpn )[[u1 , . . . , un−1 ]][u±1 ], where W is the Witt vector functor from commutative rings to commutative rings, and the generators sit in degrees |uj | = 0 and |u| = −2. This coefficient ring (or rather its formal spectrum) is a base for the universal formal deformation of the Honda formal group of height n. Example 1.1. If n = 1, then the Lubin–Tate spectrum E1 is the p–adic completion KUp of the complex topological K-theory spectrum KU. 1.2. The Morava groups. The n–th Morava stabilizer group Sn and the Galois group of Fpn over Fp both act on En such that their semi-direct product Gn , the extended Morava stabilizer group, also acts on En . Example 1.2. If n = 1, then the Morava stabilizer group G1 = S1 is the group Z× p of p–adic units which acts on E1 = KUp via Adams operations. 1.3. Devinatz–Hopkins fixed point spectra. If K Gn is a closed subgroup of the extended Morava stabilizer group Gn , then EhK n will denote the corresponding Devinatz–Hopkins fixed point spectrum [16]. For instance, in the maximal n   S (see Thm. 1(iii) of loc. cit.), as a reflection of case K = Gn , we have EhG n Morava’s change-of-rings theorem. See also [11] for a different approach. The Devinatz–Hopkins fixed point spectra are well under control in the optic of their Morava modules: There are isomorphisms hK ∼ (En )∗ (EhK n ) = π∗ (En ∧ En ) = C(Gn /K, π∗ En ),

where C(Gn /K, π∗ En ) is the ring of continuous functions from the coset space Gn /K to π∗ En with its (p, u1 , ..., un−1 )–adic topology. For the trivial group K = e this has been known to Morava (and certainly others) for a long time. See [28] for the history and a careful exposition. 1.4. Some subgroups of the Morava stabilizer group. The Morava stabilizer group acts on the Dieudonn´e module of the Honda formal group of height n, which is free of rank n over W(Fpn ). The determinant gives a homomorphism Sn → W(Fpn )× . This extends over Gn and factors through Z× p . The subgroup SGn is defined as the kernel of the (surjective) determinant, so that we have an extension 1 −→ SGn −→ Gn −→ Z× p −→ 1 of groups. Let Δ Z× p denote the torsion subgroup. If p = 2, then this subgroup is cyclic of order 2, and if p = 2, then it is cyclic of order p − 1. The pre-image of Δ under the determinant is customarily denoted by G1n . In other words, there is an extension (1.1)

1 −→ G1n −→ Gn −→ Z× p /Δ −→ 1,

and the groups SGn and G1n are then also related by a short exact sequence. 1 −→ SGn −→ G1n −→ Δ −→ 1 ∼ We remark that there are (abstract) isomorphisms Z× p /Δ = Zp of groups, but no canonical choice seems to be available.

STRING BORDISM AND CHROMATIC CHARACTERISTICS

243

1.5. The Iwasawa extensions of the local spheres. An Iwasawa extension is a (pro-)Galois extension (for instance of number fields) with Galois group isomorphic to the additive group Zp of the p–adic integers for some prime number p. The canonical Iwasawa extension of the K(n)–local sphere is the Devinatz–Hopkins fixed point spectrum hG1n

B n = En

with respect to the closed subgroup G1n . This spectrum is sometimes referred to as Mahowald’s half-sphere, in particular in the case n = 2. Example 1.3. If n = 1, then the spectrum B1 is either the 2–completion KO2 of the real topological K-theory spectrum KO (when p = 2) or the Adams summand Lp of the p–completion of the complex topological K-theory spectrum KU (when p = 2). The spectra Bn are well under control in the optic of their Morava modules: There are isomorphisms (En )∗ (Bn ) = π∗ (En ∧ Bn ) ∼ = C(Z× /Δ, π∗ En ), p

and the right hand side can be identified (non-canonically) with the ring of continuous functions on the p–adic affine line Zp . From (1.1) we infer that the spectrum Bn carries a residual action of the ∼  group Z× p /Δ = Zp , and this makes S → Bn into an Iwasawa extension of the K(n)– local sphere (see [16, bottom of p. 5]). Whenever we choose a topological generator of this group, this yields an automorphism g : Bn → Bn . Proposition 1.4. ([16, Proposition 8.1]) There is a homotopy fibration sequence g−id

(1.2)

δ

S0 −−−− → Bn −−−−→ Bn −−−−→ S1

of K(n)–local spectra. For each p and n, we fix one such fibration sequence once and for all. Example 1.5. If n = 1, then the fibration sequence g−id

(1.3)

S0 −−−→ B1 −−−→ B1

has been known for a long time. It can be extracted from [14], which in turn relies on work of Mahowald (p = 2) and Miller (p = 2). 1.6. The Hopkins–Miller classes. We are now ready to introduce the Hopkins–Miller classes ζn that are to play the role of the integral primes p in the chromatic context. Definition 1.6. The homotopy class ζn ∈ π−1  S is defined as the (de-suspension of) the composition δ

S0 −→ Bn −→ S1 of the outer maps in the homotopy fibration sequence (1.2). On the face of it, this definition seems to depend upon the choice of a topological α generator g of the group Z× p /Δ. But every other generator h has the form h = g for a p–adic unit α.

244

MARKUS SZYMIK

Lemma 1.7. For any p–adic unit α ∈ Z× p , we can write (T + 1)α − 1 = (T) · T for some unit (T) in the Iwasawa ring Zp [[T]]. Proof. Consider the function f (T) = (T + 1)α − 1 and observe that we have f (0) = 0, so that f is divisible by T. And f  (0) = α is a unit in the co efficient ring Zp by assumption. If T = g − 1, then h − 1 = g α − 1 = (T + 1)α − 1, and the lemma implies that two choices g, h of generators of Z× p /Δ yield self-maps of Bn that only differ by an equivalence. Similarly, it also makes no essential difference whether we have g − id or its negative id − g in (1.2): it changes δ (and therefore ζn ) by at most a sign. The convention in [16, §8] is different from ours. Remark 1.8. On a more conceptual level, one might be tempted to describe ζn using the canonical map S0 = (Bn )hZp → Bn → (Bn )hZp from the homotopy fixed points to the homotopy orbits, and duality. Since this point of view has, so far, not led to computational advances, we refrain from doing so. If we map S0 into the fibration sequence (1.2), then we obtain a long exact sequence (1.4)

g∗ −id

δ∗

[S0 , Bn ] −−−→ [S0 , Bn ] −−−→ [S0 , S1 ] −−−→ [S−1 , Bn ],

and ζn is, by definition, the image of the unit u under the map δ∗ . Using the defining homomorphism (1.1) of G1n , we obtain a homomorphism cn : Gn −→ Z× /Δ ∼ = Z p ⊆ π 0 En p

that we can think of as a twisted homomorphism, or 1-cocycle, and as such it defines a class in the first continuous cohomology H1 (Gn ; π0 En ). Proposition 1.9. ([16, Proposition 8.2]) The Hopkins–Miller class ζn is detected by ±cn in the K(n)–local En –based Adams–Novikov spectral sequence. The class cn is non-zero and ζn is non-zero in π−1 S0 . In fact, it generates a subgroup isomorphic to Zp in π−1 S0 . However, the class ζn becomes zero in Bn : Proposition 1.10. The composition S−1 of ζn with the unit u of Bn is zero.

ζn

/ S0

/ Bn

Proof. We have uζn = uδu, and there is already a homotopy uδ  0 as part of the fibration sequence (1.2).  Remark 1.11. The reader might wonder if perhaps π−1 Bn = 0, which is a stronger statement than the one in Proposition 1.10. But this is not always true. In fact, it is false for n = 2 and p = 2 by recent work of Beaudry, Goerss, and Henn (see [8, Cor. 8.1.6], which gives π−1 B2 = Z/2 at p = 2). However, even in that case it is still true that g∗ = id on π−1 Bn , and this allows us to prove the surjectivity of δ∗ : π0 Bn → π−1  S in (1.4) for n 2 at all primes. Injectivity follows from g∗ = id on π0 Bn , which also holds for n 2 at all primes. See [18] and [19] for the case n = 2 and p = 3. It follows that δ∗ is an isomorphism for n 2 at all primes. It appears to be open if these groups are isomorphic for n  3.

STRING BORDISM AND CHROMATIC CHARACTERISTICS

245

1.7. Chromatic characteristics. In the predecessor [48] of this paper, we have defined the notion of an E∞ ring spectrum A of prime characteristic. If p is the prime number in question, then this means that there is a null-homotopy p  0 in A. We will now work K(n)–locally and replace the prime numbers p by the Hopkins–Miller classes ζn . Definition 1.12. If A is a K(n)–local E∞ ring spectrum, and if we let uA:  S→A denote its unit, then ζn uA S −→ A ζn (A) : S−1 −→  is the associated class in π−1 A. If A is a K(n)–local E∞ ring spectrum such that there exists a null-homotopy ζn (A)  0, then we will say that A has characteristic ζn . If we write Char(ζn ) for the class of all K(n)–local E∞ ring spectra that have characteristic ζn , then we may also write A ∈ Char(ζn ) in that case. Remark 1.13. By definition, being of characteristic ζn is a property of K(n)– local E∞ ring spectra. Definition 1.12 applies more generally to K(n)–local ring spectra up to homotopy, but the examples of interest to us always come with an E∞ structure. Proposition 1.14. If A is a K(n)–local E∞ ring spectrum of characteristic ζn , then so is every K(n)–local commutative A–algebra B. Proof. The unit of any A–algebra B factors through the unit of A.



2. Chromatic and versal examples First of all, here is an example which shows that not all K(n)–local E∞ ring spectra have characteristic ζn . Example 2.1. In the initial example A =  S of the K(n)–local sphere, the unit  is the identity, so that we have ζn (S) = ζn , and this is non-zero as a consequence of Proposition 1.9. Therefore,  S ∈ Char(ζn ). This result is analogous to the fact that S ∈ Char(p) for the (un-localized) ring of spheres. Clearly, if A is a K(n)–local E∞ ring spectrum such that π−1 A vanishes, then the element ζn (A) ∈ π−1 A = 0 is automatically null-homotopic. Let us mention a couple of interesting examples of this type. Example 2.2. Because the Lubin–Tate spectra En are even spectra, we have π−1 En = 0, so that ζn (En )  0, and this implies En ∈ Char(ζn ). In other words, the Lubin–Tate spectra En all have characteristic ζn . Even if we have π−1 A = 0, or if we are perhaps in a situation when we do not know yet whether this or π−1 A = 0 holds, we might still be able to decide if ζn (A) is null-homotopic. This is the case in the following examples.

246

MARKUS SZYMIK

Example 2.3. We have Bn ∈ Char(ζn ) for all heights n by Proposition 1.10. 2.1. The versal examples. An important theoretical role in the theory of K(n)–local E∞ ring spectra of characteristic ζn is played by the versal examples. These will be introduced now. Let PX denote the free K(n)–local E∞ ring spectrum on a K(n)–local spectrum X. There is an adjunction ∼ K(n) (X, A) EK(n) ∞ (PX, A) = S between the space of K(n)–local E∞ ring maps and the space of maps of K(n)–local spectra. In one direction, the bijection sends an E∞ map PX → A to its restriction along the unit X → PX of the adjunction. (The unit of the E∞ ring spectrum PX is a map  S → PX, of course.) The inverse is denoted by x → ev(x) for any given class x : X → A. S  ζn is defined as a Definition 2.4. The K(n)–local E∞ ring spectrum  homotopy pushout PS−1 ev(ζn )

ev(0)

  S

/ S  / S  ζn

in the category of K(n)–local E∞ ring spectra. There are various ways of producing such a homotopy pushout diagram. The easiest one might be to start with a cofibrant model of  S, replacing the morphism ev(0) = P(S−1 → D0 ) by P of a cofibration S−1 → K for some contractible K, for instance the cone on S−1 , and then taking the actual pushout. See [38] for suitable notions of cofibrancy in the relevant model categories. Proposition 2.5. We have  S  ζn ∈ Char(ζn ) for all primes p. Proof. The homotopy commutativity of the enlarged diagram S−1F FF FF FF F" PS−1 ζn

 - S

0

 / S  / S  ζn

immediately shows that ζn ( S  ζn ) is homotopic to zero.



STRING BORDISM AND CHROMATIC CHARACTERISTICS

247

  ζn has the usual property Remark 2.6. The K(n)–local E∞ ring spectrum S S  ζn → A, of any homotopy pushout: a null-homotopy of ζn (A) gives rise to a map  and conversely. In fact, this allows us to add upon the preceding proposition: Any S  ζn )  0 choice of homotopy pushout  S  ζn comes with a preferred homotopy ζn ( (that corresponds to the identity map). It also implies that there is a map  (2.1) S  ζn −→ A of K(n)–local E∞ ring spectra if and only if A ∈ Char(ζn ). There is no reason why a map (2.1), once it exists, should be unique. In fact, there will usually be many such maps, even up to homotopy. This explains our use of Artin’s term ‘versal’ (from [4]) rather than ‘universal.’ Remark 2.7. As a consequence of the versal property, we have an E∞ map  S  ζn → Bn , and it is tempting to try to fit it into a pushout square  S  ζn O

/ Bn O

/ X S of K(n)–local E∞ ring spectra. Two more requirements are on my wish list for that. First, the morphism X →  S  ζn on the left is an Iwasawa extension, just like the one on the right. In particular, the spectrum X can be described as the homotopy fixed points of a Galois action on  S  ζn of a group isomorphic to Zp . Second, the spectrum X is free as an E∞ ring spectrum (X  PY for some small, K(n)–local Y ), S  ζn is adjoint to a map Y →  S  ζn of spectra, and so that the E∞ map X →  hence easier to construct. For n = 1 this rediscovers Hopkins’ cell decomposition of B1 from [22] (with Y = S0 the K(n)–local sphere). In order to demonstrate the relevance of the concept of (chromatic) characteristics outside of chromatic homotopy theory itself, we will, in the rest of this paper, give many examples of naturally occurring K(n)–local E∞ ring spectra of characteristic ζn , in particular for n = 1 and n = 2. 3. K-theories Let us start with the topological K-theory spectra. There are equivalences koK(1)  KOK(1) and kuK(1)  KUK(1) so that it is sufficient to state the results for the connective versions. Note that koK(n) and kuK(n) are contractible when n  2, so that n = 1 is the canonical height of choice. Proposition 3.1. We have koK(1) , kuK(1) ∈ Char(ζ1 ) at all primes. Proof. The K(1)–localizations (at p) agree with the p–completions of the periodic versions, compare [26, Lemma 2.3.5]. The complete periodic theories are  well known to have vanishing π−1 . The situation for algebraic K-theory spectra is different: Let Fq be an algebraic closure of a finite field Fq with q elements. If q is a power of the prime p we are working at, then the algebraic K-theory spectra K(Fq )K(1) and K(Fq )K(1) are

248

MARKUS SZYMIK

contractible by Quillen’s work [43]. (To lift his space level statements to spectra, use the Bousfield–Kuhn functor, or [39]; see [16].) We can therefore assume that the characteristic of Fq is different from p from now on, so that q ∈ Z× p . Then, again by Quillen, there is an equivalence hq

K(Fq )K(1)  E1 , and K(Fq )K(1) can be identified with the homotopy fiber E1 the self-map q − id on E1 .

of

Proposition 3.2. We have K(Fq )K(1) ∈ Char(ζ1 ) at all primes different from q. Proof. As we have remarked before, we have q ∈ Z× p , and this element has infinite order. It generates an infinite closed subgroup q such that the quotient Z× p /q hq

is finite. The long exact sequence induced by (1.3) shows that ζ1 is zero in π−1 E1 if and only if there is an element f in hq ∼ ) = C(Z× /q , Zp ) (B1 )0 (E p

1

Z× p,

such that f (gu) = f (u) + 1 for all u ∈ where g is as in (1.3). Since g has finite /q , but 1 has infinite order in Zp , such an element f order in the finite group Z× p cannot exist.  It follows immediately that many other algebraic K-theory spectra do not have characteristic ζ1 , for instance K(Z)K(1) and K(Z )K(1) for primes  = p. The same is true for K(Zp )K(1) , but this requires results of B¨okstedt–Madsen (for odd primes p) or Rognes (for p = 2). In the former case, there is a p–adic splitting K(Zp )  j ∨ Σj ∨ Σ bu, so that there is K(1)–local splitting K(Zp )  S ∨ ΣS ∨ ΣE1 , and ζ1 = 0. In the latter case, the situation is the same up to extensions: We have Σj −→ X −→ Σ ku for some spectrum X that is in X −→ K(Z2 ) −→ j (see [12] and [45]). Remark 3.3. We have been concentrating on establishing only the nontriviality of the class ζ1 for some algebraic K-theory spectra. In fact, Mitchell’s work [41] explains that several unsolved conjectures in number theory are related to the K(1)–localization of algebraic K-theory spectra, and the behavior of ζ1 on them. For instance, let F be a number field with ring OF of integers. Let  be an odd prime, and assume that OF [1/] contains the –th roots of unity. The Z –rank of π0 K(OF [1/])K(1) is the number s of primes dividing  in OF . The image of multiplication with ζ1 , ζ1 : π1 K(OF [1/])K(1) −→ π0 K(OF [1/])K(1) , lies in the Adams filtration 1 subgroup H´e2t (OF [1/]; Z (1)) of rank s − 1. It turns out that the image has maximal rank s − 1 if and only if an algebraic version of Gross’ conjecture holds (see [41, 3.6.1]).

STRING BORDISM AND CHROMATIC CHARACTERISTICS

249

Remark 3.4. Thanks to Mitchell’s earlier work [40], we know that the algebraic K-theory spectra K(R)K(n) are contractible for all (discrete) rings R and all heights n  2. This does not hold if we are willing to work with ring spectra E instead: We have ζn = 0 in K(S)K(n) , because K(S) is equivalent to Waldhausen’s A-theory of a point, and that splits off the sphere spectrum. It might be more interesting to study K(E) for E = ko or E = ku instead of E = S. See the work [5, 6] of Ausoni–Rognes. 4. Topological modular forms In this section, we discuss the spectrum tmf of topological modular forms. See the ICM talks [20, 21], [23, 24], and the Bourbaki seminar [17], for instance. Proposition 4.1. We have tmf K(2) ∈ Char(ζ2 ) at all primes. Proof. For n = 2, Behrens [9, Remark 1.7.3] has given an argument for the identification of the K(2)–localization of the spectrum of topological modular forms with EO2 , the homotopy fixed point spectrum of E2 with respect to the maximal finite subgroup M of the extended Morava group G2 , that holds for the prime p = 3. His argument can be adapted to the case p = 2 as well. Since the maximal finite subgroup M sits inside the subgroup G12 , the K(2)– localization of the topological modular forms spectrum is a commutative B2 – algebra. By Proposition 1.14 and Example 2.3, we know that T ∈ Char(ζn ) for all commutative Bn –algebras T . The situation at large primes p  5 is similar, but less well represented in the published literature. The K(2)–localization of tmf is the spectrum of global sections of the derived structure sheaf of the completion of the moduli stack of generalized elliptic curves in characteristic p at the complement of the ordinary locus. (See Behrens’ notes [10], for instance.) This sheaf can be constructed using the Goerss–Hopkins–Miller theory of Lubin–Tate spectra. The upshot is that the spectra of sections are again given by homotopy fixed points of Lubin–Tate spectra with respect to finite subgroups. These lie in the kernel of any homomorphism to a torsion-free group. (A difference is that this time their orders are co-prime to the characteristic, but this does not play a role here.) In any case, we see that the same argument as for p = 2 and p = 3 can be applied.  Remark 4.2. For n  3, we trivially have tmf K(n) ∈ Char(ζn ) as well, at all primes, because the spectrum tmf is K(n)–acyclic in that case, so that the localization vanishes. Remark 4.3. The case n = 1 is non-trivial and interesting. Hopkins has studied tmf K(1) at all primes, constructed a nullhomotopy of ζ1 on tmf K(1) and S  ζ1 with used it to describe the latter as an E∞ algebra over the versal example  one more cell attached. See [22], [36], and [10]. Remark 4.4. The K(1)–local K3 spectra from [46,47] are even hence obviously Sζ1 of characteristic ζ1 . No presentation as an E∞ algebra over the versal example  is known in these cases.

250

MARKUS SZYMIK

Remark 4.5. The strategy for the proof of Proposition 4.1 can also be pursued to show that the higher real K-theories EOp−1 have chromatic characteristics. 5. Bordism theories Since the sphere spectrum represents framed bordism, it is clear that not all bordism spectra have chromatic characteristics. In this section we discuss the bordism spectra MSpin and MString as well as their complex cousins MSU and MU6 . Proposition 5.1. We have MSpinK(n) ∈ Char(ζn ) at all primes and all heights n  1. Proof. At odd primes, the spectrum MSpin is equivalent to a wedge of even suspensions of BP. (The splitting as a wedge of Brown–Peterson spectra is wellknown [26]. It can be deduced from one of Steinberger’s general splitting results [15, Theorem III.4.3]. We can then work rationally in order to see that only even suspensions are necessary. And rationally, both MSpin and BP are even. Compare a similar argument in the proof of Proposition 5.4 below.) Consequently, the K(n)– localizations of MSpin are well understood at odd primes. The K(n)–localization of BP has π∗ BPK(n) = (vn−1 π∗ BP)p,v1 ,...,vn−1 . This has been explained by Hovey [25, Lemma 2.3], for instance. We see that the homotopy groups π∗ BPK(n) are concentrated in even degrees. This clearly implies π−1 = 0 for the K(n)–localizations of MSpin, and a fortiori these have characteristic ζn . The even prime p = 2 affords some extra arguments. Since the Spin bordism spectrum MSpin, as any Thom spectrum, is connective, we in particular have π−1 MSpin = 0. This does not imply the result for the K(1)–localization, however (think of S). But, the Anderson–Brown–Peterson (ABP) splitting shows that this still holds after K(1)–localization, since the spectrum MSpin splits K(1)– locally at the prime p = 2 as a wedge of (unsuspended) localizations of copies of the spectrum ko (compare [26, Proposition 2.3.6]). ⎛ ⎞ % MSpinK(1)  ⎝ KO⎠ j

2

Therefore, the result follows from what we have said for the K-theories in Section 3, Proposition 3.1. For heights n  2, the spectrum MSpin is K(n)–acyclic, again by the ABP splitting.  Remark 5.2. At odd primes, the spectrum MSU also decomposes into even suspensions of BP. (This time, the splitting is explicitly stated by Steinberger [15, Remarks III.4.4], and ‘even’ follows again by rational considerations.) We can similarly conclude that MSUK(n) ∈ Char(ζn ) at odd primes and all heights n  1. At the prime p = 2, the situation is substantially different, since no simple ABP-type splitting is known. See Pengelley [42], who found the BoP summands. According to Reeker’s thesis [44], we have at least MSUK(1) ∈ Char(ζ1 ).

STRING BORDISM AND CHROMATIC CHARACTERISTICS

251

The canonical maps MString → MSpin and MU6 → MSU of bordism spectra are both K(1)–local equivalences [26, Prop. 2.3.1]. Therefore, we immediately get: Corollary 5.3. We have MStringK(1) , MU6 K(1) ∈ Char(ζ1 ) at all primes p. For n  2 we can offer the following result. Proposition 5.4. We have MStringK(n) , MU6 K(n) ∈ Char(ζn ) at all primes p  5 and all heights n  2. Proof. If p  5, then both MString and MU6 split p–locally as wedges of suspensions of BP by [29, Corollary 2.2]. See also [27]. % Σmj BP(p) MU6 (p)  j

MString(p) 

%

Σnj BP(p)

j

We can work rationally in order to obtain information about the suspensions mj and nj needed, and that is easy: Since π∗ MU6 Q ∼ = Q[c2 , c3 , . . . ] with |cn | = 2n, n , v , . . . ] with |v | = 2(p − 1), we see that the mj are even. Q[v and π∗ BPQ ∼ = 1 2 n Similarly for the nj , using π∗ MStringQ ∼ = Q[p2 , p3 , . . . ] and |pn | = 4n. A fortiori, these additive decompositions exist also K(n)–locally at the prime in question. The K(n)–localization of BP has (5.1)

π∗ BPK(n) = (vn−1 π∗ BP)p,v1 ,...,vn−1 ,

see [25, Lemma 2.3] again. Therefore, the homotopy groups of both of the spectra MU6 K(n) and MStringK(n) are concentrated in even degrees. We can deduce that both of the groups π−1 MStringK(n) and π−1 MU6 K(n) vanish for primes p  5, from which the statement follows.  Remark 5.5. For the small primes p = 2 and p = 3 it is still true that we are able to find finite complexes F (depending on p) with cells only in even dimensions such that MString ∧ F and MU6 ∧ F split as wedges of (even) suspensions of BP (see [29, Corollary 2.2]). Strictly speaking, this excludes the case MString at the prime p = 2. But, since there is a map MU6 −→ MString, Proposition 1.14 guarantees that it would be sufficient to prove that MU6 has chromatic characteristics to be able to infer that for the string bordism spectrum as well. Acknowledgments I thank H.-W. Henn, E.C. Peterson, and A. Salch for informative conversations, and the referees for their detailed and helpful reports. This research has been supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).

252

MARKUS SZYMIK

References [1] M. Ando, M.J. Hopkins, C. Rezk. Multiplicative orientatons of KO–theory and the spectrum of topological modular forms. Preprint. www.math.uiuc.edu/~mando/papers/koandtmf.pdf [2] M. Ando, M. J. Hopkins, and N. P. Strickland, Elliptic spectra, the Witten genus and the theorem of the cube, Invent. Math. 146 (2001), no. 3, 595–687, DOI 10.1007/s002220100175. MR1869850 [3] M. Ando, M. J. Hopkins, and N. P. Strickland, The sigma orientation is an H∞ map, Amer. J. Math. 126 (2004), no. 2, 247–334. MR2045503 [4] M. Artin, Algebraization of formal moduli. I, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 21–71. MR0260746 [5] C. Ausoni, On the algebraic K-theory of the complex K-theory spectrum, Invent. Math. 180 (2010), no. 3, 611–668, DOI 10.1007/s00222-010-0239-x. MR2609252 [6] C. Ausoni and J. Rognes, Algebraic K-theory of topological K-theory, Acta Math. 188 (2002), no. 1, 1–39, DOI 10.1007/BF02392794. MR1947457 [7] A. Baker, Characteristics for E∞ ring spectra, An alpine bouquet of algebraic topology, Contemp. Math., vol. 708, Amer. Math. Soc., Providence, RI, 2018, pp. 1–17, DOI 10.1090/conm/708/14265. MR3807749 [8] A. Beaudry, P.G. Goerss, H.-W. Henn. Chromatic splitting for the K(2)-local sphere at p = 2. Preprint. arxiv.org/abs/1712.08182 [9] M. Behrens, A modular description of the K(2)-local sphere at the prime 3, Topology 45 (2006), no. 2, 343–402, DOI 10.1016/j.top.2005.08.005. MR2193339 [10] M. Behrens. The construction of tmf. Topological modular forms (Talbot Workshop 2007) 131–188. Mathematical Surveys and Monographs 201. American Mathematical Society, 2014. [11] M. Behrens and D. G. Davis, The homotopy fixed point spectra of profinite Galois extensions, Trans. Amer. Math. Soc. 362 (2010), no. 9, 4983–5042, DOI 10.1090/S0002-9947-10-05154-8. MR2645058 [12] M. B¨ okstedt and I. Madsen, Topological cyclic homology of the integers, Ast´ erisque 226 (1994), 7–8, 57–143. K-theory (Strasbourg, 1992). MR1317117 [13] B. I. Botvinnik, Structure of the ring M SU∗ (Russian), Mat. Sb. 181 (1990), no. 4, 540– 555, DOI 10.1070/SM1991v069n02ABEH001377; English transl., Math. USSR-Sb. 69 (1991), no. 2, 581–596. MR1055528 [14] A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), no. 4, 257–281, DOI 10.1016/0040-9383(79)90018-1. MR551009 [15] R. R. Bruner, J. P. May, J. E. McClure, and M. Steinberger, H∞ ring spectra and their applications, Lecture Notes in Mathematics, vol. 1176, Springer-Verlag, Berlin, 1986. MR836132 [16] E. S. Devinatz and M. J. Hopkins, Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups, Topology 43 (2004), no. 1, 1–47, DOI 10.1016/S00409383(03)00029-6. MR2030586 [17] P. G. Goerss, Topological modular forms [after Hopkins, Miller and Lurie], Ast´ erisque 332 (2010), Exp. No. 1005, viii, 221–255. S´ eminaire Bourbaki. Volume 2008/2009. Expos´es 997– 1011. MR2648680 [18] P. G. Goerss, H.-W. Henn, and M. Mahowald, The rational homotopy of the K(2)-local sphere and the chromatic splitting conjecture for the prime 3 and level 2, Doc. Math. 19 (2014), 1271–1290. MR3312144 [19] P. Goerss, H.-W. Henn, M. Mahowald, and C. Rezk, On Hopkins’ Picard groups for the prime 3 and chromatic level 2, J. Topol. 8 (2015), no. 1, 267–294, DOI 10.1112/jtopol/jtu024. MR3335255 [20] M. J. Hopkins, Topological modular forms, the Witten genus, and the theorem of the cube, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Z¨ urich, 1994), Birkh¨ auser, Basel, 1995, pp. 554–565. MR1403956 [21] M. J. Hopkins, Algebraic topology and modular forms, Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 291–317. MR1989190 [22] M. J. Hopkins, K(1)-local E∞ -ring spectra, Topological modular forms, Math. Surveys Monogr., vol. 201, Amer. Math. Soc., Providence, RI, 2014, pp. 287–302, DOI 10.1090/surv/201/16. MR3328537

STRING BORDISM AND CHROMATIC CHARACTERISTICS

253

[23] M. J. Hopkins and M. Mahowald, From elliptic curves to homotopy theory, Topological modular forms, Math. Surveys Monogr., vol. 201, Amer. Math. Soc., Providence, RI, 2014, pp. 261– 285, DOI 10.1090/surv/201/15. MR3328536 [24] M. J. Hopkins and H. R. Miller, Elliptic curves and stable homotopy I, Topological modular forms, Math. Surveys Monogr., vol. 201, Amer. Math. Soc., Providence, RI, 2014, pp. 209–260, DOI 10.1090/surv/201/14. MR3328535 [25] M. Hovey, Bousfield localization functors and Hopkins’ chromatic splitting conjecture, The ˇ Cech centennial (Boston, MA, 1993), Contemp. Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995, pp. 225–250, DOI 10.1090/conm/181/02036. MR1320994 [26] M. A. Hovey, vn -elements in ring spectra and applications to bordism theory, Duke Math. J. 88 (1997), no. 2, 327–356, DOI 10.1215/S0012-7094-97-08813-X. MR1455523 [27] M. Hovey, The homotopy of M String and M U6 at large primes, Algebr. Geom. Topol. 8 (2008), no. 4, 2401–2414, DOI 10.2140/agt.2008.8.2401. MR2465746 [28] M. Hovey, Operations and co-operations in Morava E-theory, Homology Homotopy Appl. 6 (2004), no. 1, 201–236. MR2076002 [29] M. A. Hovey and D. C. Ravenel, The 7-connected cobordism ring at p = 3, Trans. Amer. Math. Soc. 347 (1995), no. 9, 3473–3502, DOI 10.2307/2155020. MR1297530 [30] M. Hovey and N. P. Strickland, Morava K-theories and localisation, Mem. Amer. Math. Soc. 139 (1999), no. 666, viii+100, DOI 10.1090/memo/0666. MR1601906 [31] M. Joachim, A symmetric ring spectrum representing KO-theory, Topology 40 (2001), no. 2, 299–308, DOI 10.1016/S0040-9383(99)00063-4. MR1808222 [32] M. Joachim, Higher coherences for equivariant K-theory, Structured ring spectra, London Math. Soc. Lecture Note Ser., vol. 315, Cambridge Univ. Press, Cambridge, 2004, pp. 87–114, DOI 10.1017/CBO9780511529955.006. MR2122155 [33] S. O. Kochman, The ring structure of BoP∗ , Algebraic topology (Oaxtepec, 1991), Contemp. Math., vol. 146, Amer. Math. Soc., Providence, RI, 1993, pp. 171–197, DOI 10.1090/conm/146/01222. MR1224914 [34] G. Laures, The topological q-expansion principle, Topology 38 (1999), no. 2, 387–425, DOI 10.1016/S0040-9383(98)00019-6. MR1660325 [35] G. Laures, An E∞ splitting of spin bordism, Amer. J. Math. 125 (2003), no. 5, 977–1027. MR2004426 [36] G. Laures, K(1)-local topological modular forms, Invent. Math. 157 (2004), no. 2, 371–403, DOI 10.1007/s00222-003-0355-y. MR2076927 [37] L. G. Lewis Jr., J. P. May, M. Steinberger, and J. E. McClure, Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure. MR866482 [38] M. A. Mandell, J. P. May, S. Schwede, and B. Shipley, Model categories of diagram spectra, Proc. London Math. Soc. (3) 82 (2001), no. 2, 441–512, DOI 10.1112/S0024611501012692. MR1806878 [39] J. P. May, E∞ ring spaces and E∞ ring spectra, Lecture Notes in Mathematics, Vol. 577, Springer-Verlag, Berlin-New York, 1977. With contributions by Frank Quinn, Nigel Ray, and Jørgen Tornehave. MR0494077 [40] S. A. Mitchell, The Morava K-theory of algebraic K-theory spectra, K-Theory 3 (1990), no. 6, 607–626, DOI 10.1007/BF01054453. MR1071898 [41] S. A. Mitchell, K(1)-local homotopy theory, Iwasawa theory and algebraic K-theory, Handbook of K-theory. Vol. 1, 2, Springer, Berlin, 2005, pp. 955–1010, DOI 10.1007/978-3-54027855-9 19. MR2181837 [42] D. J. Pengelley, The homotopy type of M SU, Amer. J. Math. 104 (1982), no. 5, 1101–1123, DOI 10.2307/2374085. MR675311 [43] D. Quillen, On the cohomology and K-theory of the general linear groups over a finite field, Ann. of Math. (2) 96 (1972), 552–586, DOI 10.2307/1970825. MR0315016 [44] H. Reeker. On K(1)–local SU–bordism. Dissertation, Bochum, 2009. [45] J. Rognes, Algebraic K-theory of the two-adic integers, J. Pure Appl. Algebra 134 (1999), no. 3, 287–326, DOI 10.1016/S0022-4049(97)00156-4. MR1663391 [46] M. Szymik, K3 spectra, Bull. Lond. Math. Soc. 42 (2010), no. 1, 137–148, DOI 10.1112/blms/bdp106. MR2586974 [47] M. Szymik, Crystals and derived local moduli for ordinary K3 surfaces, Adv. Math. 228 (2011), no. 1, 1–21, DOI 10.1016/j.aim.2011.05.004. MR2822225

254

MARKUS SZYMIK

[48] M. Szymik, Commutative S-algebras of prime characteristics and applications to unoriented bordism, Algebr. Geom. Topol. 14 (2014), no. 6, 3717–3743, DOI 10.2140/agt.2014.14.3717. MR3302977 Department of Mathematical Sciences, NTNU Norwegian University of Science and Technology, 7491 Trondheim, Norway Email address: [email protected]

Contemporary Mathematics Volume 729, 2019 https://doi.org/10.1090/conm/729/14699

Mahowald square and Adams differentials Zhouli Xu Abstract. This article is a survey on the RP ∞ -method that was used to compute differentials in the Adams spectral sequence of the sphere spectrum. This method was introduced in [Ann. of Math. (2) 186 (2017), no. 2, 501–580] by Guozhen Wang and the author and was also used in [Algebr. Geom. Topol. 18 (2018), no. 7, 3887–3906] to solve extension problems in the Adams spectral sequence. The method is based on the algebraic Kahn-Priddy theorem and the Mahowald square. In this article, we discuss the idea of the RP ∞ -method and apply this method to prove certain Adams differentials in low stems. We also discuss a way of constructing Toda brackets for elements in the stable homotopy groups of sphere using the Kahn-Priddy transfer map.

Contents 1. Introduction 2. The mod 2 Moore spectrum M at stem 14—a warmup 3. RP ∞ -method for some Adams differentials in low stems 4. The Kahn-Priddy map and Toda brackets References

1. Introduction At the prime 2, the Adams spectral sequence is still the most effective tool for computing stable stems. The E2 -page of the Adams spectral sequence is comparably tractable—one may use the May spectral sequence, the Curtis algorithm for Lambda algebras, or computer programs such as the ones made by Bob Bruner, Christian Nassau and Amelia Perry to compute it in a large range. The more difficult part is to compute Adams differentials and to solve extension problems. Many methods have been developed to compute Adams differentials. Historically, up to the 19-stem, the differentials are obtained by comparing with Toda’s unstable computations—one deduces the differentials by knowing what to expect in the E∞ -page. One could also deduce these differentials using the solution of the Hopf invariant one problem [1] and computations of the image-of-J spectrum [7, 16]. Up to the 28-stem, the differentials are obtained by May [21] using the multiplicative structure based on earlier differentials. The range from 30 to 45 is due to Barratt-Mahowald-Tangora [3]. They have two major techniques. The first one is Moss’s theorem [22], which connects Massey products on any Adams Er page to Toda brackets in homotopy. The second one is a trick due to Mahowald, c 2019 American Mathematical Society

255

256

ZHOULI XU

which translates differentials and extensions through certain finite CW complexes. One can generalize Mahowald’s trick to the Mahowald square in the setting of the 4 spectral sequences situation—a combination of an Atiyah-Hirzebruch spectral sequence (AHSS), an algebraic Atiyah-Hirzebruch spectral sequence (AAHSS) and a (cellwise) Adams spectral sequence (ASS). This general method has been used by many experts for various spectra. See Mahowald’s work [17] on the metastable computation, for example. Another method [4] involved in this range is Bruner’s use of Steenrod operations in the ASS. The Adams differentials in the range up to 59 have been proved rigorously by Isaksen [10–12] using motivic analogues of the classical methods. Recently, Gheorghe, Isaksen, Wang and the author [9, 13] have developed another method through the motivic world that computes Adams differentials using algebraic Novikov differentials. The computation has been pushed up to the 90 stem, but is orthogonal to the theme of this article. Recently, based on the algebraic Kahn-Priddy theorem [15] and Mahowald’s trick, Guozhen Wang and the author [26] introduced the RP ∞ -method. The method has been applied to prove a notoriously hard Adams differential d3 (D3 ) = B3 in stem 61 (see [26]), which is one major reason why Isaksen’s computations [10] stopped at stem 59. The method has also been applied to solve certain extension problems (see [28]), which include the last 2-extension problem that was left by Isaksen up to stem 59. The starting point of the RP ∞ -method is the following algebraic version of the Kahn-Priddy theorem which is due to W. H. Lin [15]. Theorem 1.1. Let t : P1∞ → S 0 be the transfer map, where P1∞ is the suspension spectrum of RP ∞ and S 0 is the sphere spectrum. Then the induced map on the Adams E2 -page s+1,t+1 ∞ t : Exts,t (Z/2, Z/2) A (Z/2, H∗ (P1 )) −→ ExtA

is an epimorphism for bidegrees such that t − s > 0. Here A is the dual Steenrod algebra. The observation we made is that according to the (algebraic) Kahn-Priddy theorem [14,15], if there is a certain nontrivial dr differential in the Adams spectral sequence for the sphere spectrum, then the pre-image of the source must support a nontrivial dr differential for r  ≤ r and the pre-image of the target cannot be killed by a nontrivial dr differential for r  < r. Therefore, to compute nontrivial Adams differentials for the sphere spectrum, we can work with certain Adams differentials for P1∞ and use the Mahowald square to compute the differentials. Here both the AHSS and the AAHSS come from the skeletal filtration on P1∞ , and they both start from the E1 -page. The d1 -differentials in both spectral sequences correspond to a multiplication map by 2 or 0, depending on the dimension of the cells. In theory, the longer differentials in the AAHSS can be computed by (matrix) Massey products and the longer differentials in the AHSS can be computed by (matrix) Toda brackets. In fact, in [27], Wang and the author computed the Adams E2 -page of P1∞ in the range of t < 72 using the Lambda algebra. This Lambda algebra computation gives us a lot of information on the algebraic AtiyahHirzebruch spectral sequence. In particular, there is a one-to-one correspondence between the differentials in the Lambda algebra computation and the differentials in

MAHOWALD SQUARE AND ADAMS DIFFERENTIALS

257

the algebraic Atiyah-Hirzebruch spectral sequence. From the Lambda algebra comn+k )), where putation, one can also read off information about Ext∗,∗ A (Z/2, H∗ (Pn n+k n+k n−1 is the suspension spectrum of RP /RP . We also computed the map Pn t in the same range. The AHSS differentials on the other hand are much harder to compute. Toda π∗ P1∞ ; 3 i a L brackets L p L LLLLL pp LLLLLL ppp p p LLLLLL p p p Atiyah-Hirzebruch LLLL SS Adams SS p LLLLL ppp p p LLLLLL p p p LLLL pp ∞ ∗,∗ ∞ ExtA (Z/2, H∗ (P1 )) π∗ (S n ) n=1 ck NN = 5 s sss NNNN ssssss NNNN s s s s N ssssss algebraic Atiyah-Hirzebruch SS NNN cellwises Adams SS s s NNNN ssss sssss s NNN s s s N ssss ∞ Massey ∗,∗ Ext (Z/2, H∗ (S n )) A n=1 products

Mahowald square Now assuming we have enough information in the AAHSS, AHSS and the cellwise ASS, we can deduce differentials in the ASS by comparison. In practice, since ∞ the AAHSS computation tells us which cell in P1∞ a class in Ext∗,∗ A (Z/2, H∗ (P1 )) ∞ comes from, we could work with an HF2 -subquotient of P1 . Here an HF2 subcomplex of P1∞ is a CW spectrum together with a map to P1∞ that induces a monomorphism on mod 2 homology and an HF2 -quotient complex of P1∞ is a CW spectrum together with a map from P1∞ that induces an epimorphism on mod 2 homology. An HF2 -subquotient means a “zigzag” of maps between spectra that induce a “zigzag” of monomorphisms and epimorphisms on mod 2 homology. (See Definition 4.1 of [26] for a precise definition.) Comparing with P1∞ , its HF2 subquotient has the advantage of having fewer cells, chosen depending on the source and target of a proposed Adams differential. Therefore, it is easier to compute its Atiyah-Hirzebruch differentials. Ideally, we can keep track of Adams differentials in P1∞ through certain HF2 -subquotients with as few cells as possible, establish the Adams differential in the final HF2 -subquotient, and use naturality of Adams spectral sequences to deduce the differential in the ASS for the sphere. The crucial part of the RP ∞ -method is to know which Adams differentials and which spectra are to be considered. This is often a case-by-case situation and is usually suggested by a “zigzag” of differentials in the AAHSS, AHSS and the cellwise ASS. The idea of the “zigzag” process can be found, for example, in [8]. Note that the “zigzag” process only provides intuition—the rigorous proof has to be made using naturality of spectral sequences with careful checking of filtration jumps. We will give an example in Section 2. Although the RP ∞ -method is complicated and ad hoc, it is actually inductive in the following sense. Suppose that the source of an Adams differential in the

258

ZHOULI XU

sphere spectrum is in stem n. The pre-image of the source in stem n is a class in stem n for P1∞ , which must come from a class on a cell in positive dimension in the E1 -page of the AAHSS, since P1∞ has cells in only positive dimensions. The internal stem of this class is in a stem strictly smaller than n. Therefore, if the ASS of all HF2 -subquotients of P1∞ are well understood up to stem n, then one should be able to deduce Adams differentials for the sphere spectrum in stem n + 1. Similar ideas apply to extension problems as well. The Kahn-Priddy map is also helpful in constructing Toda brackets. It is a theorem due to Cohen [6] that every element in the stable homotopy groups of spheres can be decomposed as Toda brackets using only 2, η, ν and σ, at the prime 2. Note that in Cohen’s proof, the Toda brackets really mean mapping through certain finite cell complexes, in which all attaching maps can be constructed by iteration of Toda brackets of the Hopf maps. It does not necessarily give us a matric Toda bracket where all intermediate complexes are wedges of spheres. Nevertheless, in practice, there is no universal way of constructing such a Toda bracket for a given class in stable stems. The first element that the author does not know how to express in this form is the class θ4.5 defined by Isaksen [10] in stem 45, which is detected by h34 in the ASS. Knowing a (matric) Toda bracket is in general very useful in computations of stable stems. Since many classes in the range after 45 are divisible by θ4.5 , knowing a Toda bracket of it would be very helpful in proving certain relations and differentials in the ASS. Suppose we have a class α in stem n. Then the Kahn-Priddy theorem tells us it maps through P1∞ . Moreover, it must map through P1n . All primary attaching maps in P1∞ are well understood—they are in fact detected by the J-spectrum. The Kahn-Priddy map is well understood as well—when restricting to P1n , the cofiber is a desuspension of an HF2 -subquotient of P1∞ due to James periodicity. Therefore, the resulting composite Sn

/ P1n

/ S0

realizing α could be viewed as a matric Toda bracket in a generalized sense. This again does not give us a matric Toda bracket where all intermediate complexes are wedges of spheres, but it is practical to get such a decomposition. We will give a few examples in Section 4 including a tentative one for the class θ4.5 in π45 . The rest of the article is organized as follows. In Section 2, we will review Mahowald’s trick on translation between differentials and extensions through an example—the mod 2 Moore spectrum at stem 14. In Section 3, we prove certain Adams differentials in low stems using the RP ∞ -method. We will put emphasis on the “zigzag” process that predicts Adams differentials and HF2 -subquotients. In Section 4, we give a few constructions of Toda brackets using the Kahn-Priddy map. Acknowledgment. The author thanks the anonymous referee for helpful suggestions on the draft of this survey article. 2. The mod 2 Moore spectrum M at stem 14—a warmup The first few nontrivial Adams differentials for the sphere spectrum are in stems 14 and 15. Namely, d2 (h4 ) = h0 h23 , d3 (h0 h4 ) = h0 d0 , d3 (h20 h4 ) = h20 d0 .

MAHOWALD SQUARE AND ADAMS DIFFERENTIALS

259

Knowing these differentials, there are two elements left in the E∞ -page of the ASS: h23 and d0 . So there are two possibilities for π14 : Z/2 ⊕ Z/2 or Z/4, depending on a possible 2-extension from h23 to d0 . For stem 15, there is a possible d3 differential from h1 h4 to h1 d0 . The purpose of this section is to show the following two statements are equivalent using the mod 2 Moore spectrum M . (1) There is a nontrivial 2-extension in the 14-stem from h23 to d0 . (2) There is a nontrivial differential d3 (h1 h4 ) = h1 d0 . We start with the Adams E2 -page of M . The cofiber sequence S0

/M

/ S1

gives us a short exact sequence on homology with Z/2-coefficients / H∗ (S 0 )

0

/ H∗ (M )

/ H∗ (S 1 )

/0

and therefore a long exact sequence on Ext groups ···

/ Exts,t (Z/2, Z/2) A

/ Exts,t (Z/2, H∗ M ) A

/ Exts,t−1 (Z/2, Z/2) A

/ ···

where the boundary homomorphism is multiplication by h0 : h0 : Exts,t−1 (Z/2, Z/2) −→ Exts+1,t (Z/2, Z/2). A A We also have the long exact sequence on homotopy groups with boundary homomorphism multiplication by 2, which is compatible with the long exact sequence of the Ext groups—this is the naturality of the ASS. ···

/ Exts,t (Z/2, Z/2) A

/ Exts,t (Z/2, H∗ M ) A

/ Exts,t−1 (Z/2, Z/2) A

/ ···

···

 / πt−s (S 0 )

 / πt−s (M )

 / πt−s (S 1 )

/ ···

The long exact sequence on homotopy groups can be viewed as the AHSS for M , while the long exact sequence of the Ext groups can be viewed as the AAHSS for M . To fix notation, for any classes in π∗ (M ) and Ext∗,∗ A (Z/2, H∗ M ), we denote them by α[n] and a[n], as the elements in the E1 -page of the (algebraic) AHSS detecting them, where α ∈ π∗ (S 0 ), a ∈ Ext∗,∗ A (Z/2, Z/2) and n is the grading from the cellular filtration—so either 0 or 1. In the next page, we have a chart of three spectral sequences for S 0 and M at stems 14-16. The one on the left is the ASS of S 0 . There are three nontrivial differentials in this range. The one in the middle is the AAHSS for M . All AAHSS differentials in this range are d1 -differentials. The one on the right consists of the remaining dots in the middle AAHSS, which is the E2 -page of the ASS for M . It turns out there are no differentials in this range, so it is also the E∞ -page of the ASS for M .

260

ZHOULI XU

The Adams spectral sequences of S 0 and M at stems 14-16 8

4

8

4

8

4 P c0 [0]

P c0

6

3

6

3

6

3 h2 0 d0 [1]

h1 d0 [1]

4

2

4

d0

2

4

2 d0 [0]

h0 h2 3 [1]

2

1

2

h2 3

1

2

1 h2 3 [0]

h4

0

h4 [0]

0 0 14

1 16

0

0 2 14

3 16

0

0 4 14

5 16

Now suppose that there is a nontrivial 2-extension from h23 to d0 . This is the same thing as saying there is a nontrivial Atiyah-Hirzebruch d1 -differential that is not seen in the AAHSS. Therefore π14 (M ) must be Z/2 generated by σ 2 [0] by the long exact sequence on homotopy groups. For the ASS of M , the only possibility is a d3 -differential: d3 (h4 [0]) = d0 [0]. By the Leibniz rule of h1 -multiplication, we must have d3 (h1 h4 [0]) = h1 d0 [0]. This further implies that d3 (h1 h4 ) = h1 d0 in the ASS for the sphere spectrum by naturality. Since each step of the above argument can be reversed, it follows that the two statements are equivalent. Admittedly, there are other ways to see the two statements are equivalent and to prove the contrary of each statement is true. For example, if d0 detects a class which is divisible by 2, then h1 d0 must be killed since 2η = 0 and h1 h4 is the only possibility. This shows that (1) implies (2). For the other implication, using the differential d2 (h4 ) = h0 h23 , we have a Massey product in the Adams E3 -page: h1 h4 = h23 , h0 , h1 .

MAHOWALD SQUARE AND ADAMS DIFFERENTIALS

261

Then Moss’s theorem tells us if 2 · σ 2 = 0 then h1 h4 is a permanent cycle. This shows that (2) implies (1). For statement (1), since σ is in an odd stem, and the stable homotopy ring of spheres is commutative in the graded sense, then we must have 2 · σ 2 = 0. For statement (2), Mahowald [18] in fact proved that h1 hj survives for all j ≥ 3. One could also use the Hurewicz image of tmf to show that κ (the homotopy class detected by d0 ) is not divisible by 2 and that ηκ is nonzero. It is interesting to compare classes in the E∞ -page of the ASS of M and classes in the E∞ -page of the AHSS of M . Let ρ15 be a homotopy class in π15 that is detected by h30 h4 . Since there are only d1 -differentials in the AHSS for M , it is clear that the 15-stem of the E∞ -page of the AHSS of M is generated by ηκ[0], ρ15 [0], σ 2 [1], κ[1]. The 15-stem of the E∞ -page of the ASS of M is generated by h4 [0], h0 h23 [1], h1 d0 [0], h20 d0 [1]. To see the correspondence, we compare the Adams filtrations. The class σ 2 [1] has Adams filtration 2 in S 1 , so its Adams filtration in M is at most 2. The only possibility is that h4 [0] detects σ 2 [1]. Similarly, the class κ[1] is detected by h0 h23 [1], the class ηκ[0] is detected by h1 d0 [0], and the class ρ15 [0] is detected by h20 d0 [1]. Knowing the correspondence from the naturality of the ASS, we explain how to use the “zigzag” process to predict the same answer. Start with the class σ 2 [1]: it is supposed to be detected by h23 [1]. However, h23 [1] supports an algebraic AtiyahHirzebruch differential that kills h0 h23 [0] so it is not present on the Adams E2 -page. Furthermore, the element h0 h23 [0] is killed by h4 [0] in the cellwise ASS. In summary, we have two differentials d1 (h23 [1]) = h0 h23 [0] in the AAHSS, d2 (h4 [0]) = h0 h23 [0] in the cellwise ASS. Combining them together, it leads us to deduce that h4 [0] detects σ 2 [1]. As another example, the fact that the class κ[1] is detected by h0 h23 [1] follows from the following “zigzag” of 4 differentials: d1 (d0 [1]) = h0 d0 [0] in the AAHSS, d3 (h0 h4 [0]) = h0 d0 [0] in the cellwise ASS, d1 (h4 [1]) = h0 h4 [0] in the AAHSS, d2 (h4 [1]) = h0 h23 [1] in the cellwise ASS. As a final comment on this computation, we actually have π15 (M ) = Z/4 ⊕ Z/2 ⊕ Z/2. The class h21 d0 [0] is killed by a d2 -differential and there is a nontrivial 2-extension from h0 h23 [1] to h1 d0 [0]. It is easier to see this 2-extension from the AHSS. We have 2 · κ[1] = 2, κ, 2 [0] = ηκ[0]. In general, suppose that α · β = 0, β · γ = 0. Then in the homotopy of the cofiber of α, we always have a γ-extension in terms of the Atiyah-Hirzebruch notations: γ · β[|α| + 1] = α, β, γ [0]. Here |α| is the degree of α. (See, for example, Lemma 5.2 in [28] for a proof.)

262

ZHOULI XU

3. RP ∞ -method for some Adams differentials in low stems We start with commenting on the Hopf invariant one differential d2 (h4 ) = h0 h23 . Since it is a d2 -differential, there is only one possibility—the pullback differential in P1∞ has to be a d2 -differential as well. Following computations in [28], we see that the differential in P1∞ is d2 (1[15]) = h1 h3 [6]. This leads us to consider a certain HF2 -subquotient of P1∞ that contains both the 15-cell and the 6-cell. In fact, by the solution of the Hopf invariant one problem, we know that the primary attaching map from the 15-cell goes down to the 6-cell by ησ ∈ π8 . In other words, we have a splitting of finite spectra P715  P714 ∨ S 15 , and Σ6 C(ησ) is an HF2 -subcomplex of P615 :  / P615 , Σ6 C(ησ)  where C(ησ) is the cofiber of ησ. Therefore, the Adams d2 -differential is compatible with the Atiyah-Hirzebruch differential d9 (1[15]) = ησ[6]. One can first use the Atiyah-Hirzebruch differential to prove an Adams d2 -differential in Σ6 C(ησ) and then use naturality to prove the same differential in P1∞ and therefore the one in the sphere. Of course this “proof” is circular due to the Hopf invariant one problem, but it is nice to see the connection through the Mahowald square. In the rest of this section, we give a proof of the Adams differential in the 30-stem d3 (r) = h1 d20 by the RP ∞ -method. We choose this differential for the following reason: up to the 45-stem, this is essentially the only Adams differential that cannot be computed by the recent motivic method. It can, however, be computed by Bruner’s power operation arguments or by an ad hoc argument due to Mahowald-Tangora [19], for example. Since d0 is a permanent cycle, it is equivalent to give a proof of the Adams differential in the 44-stem: d3 (d0 r) = h1 d30 . Checking computations in [28], we see the pre-images in the ASS for P1∞ of d0 r and h1 d30 are v[2] and d30 [1]. This is a great sign that the cells involved in P1∞ are in very low dimensions. Now let’s do the “zigzag”: d2 (v[2]) = h21 u[2] in the cellwise ASS, d2 (h1 u[4]) = h21 u[2] in the AAHSS. The class h1 u is a surviving cycle in π40 and detects 2κ2 . The minimal HF2 subcomplex of P1∞ that contains the 4-cell is P14 , and it has an HF2 -quotient complex P14 /S 3 since S 3 is an HF2 -subcomplex of P14 . The attaching maps in P14 /S 3 are η and 2. Therefore, in the AHSS for both P14 /S 3 and P14 , the class 2κ2 [4] supports a differential: d3 (2κ2 [4]) = 2κ, η, 2 [1] = η 2 κ2 [1].

MAHOWALD SQUARE AND ADAMS DIFFERENTIALS

263

This is great since η 2 κ2 is exactly detected by d30 ! This completes the “zigzag” process. Now we move on to a rigorous proof using naturality. What we need is to prove a nontrivial Adams differential in P1∞ : d3 (v[2]) = d30 [1]. Since the “zigzag” of differentials are among elements from cells in dimensions between 1 and 4, checking computations in [28], we have the Adams E2 -page of P14 in stems 43 and 44. One could also easily do the AAHSS computation by hand. Table 1. The Adams E2 -page of P14 and P34 in the 43 and 44stems for s ≤ 12

s\t − s 12 11 10

4 Exts,t A (Z/2, H∗ (P1 )) 43 44 d30 [1] • • • z[3]

9 8 7 6

• • •

5 4

4 Exts,t A (Z/2, H∗ (P3 )) 44

z[3] h1 u[4]

v[2]

h1 f1 [2] • •

3

g 2 [4] h1 P h1 h5 [3] h1 P h1 h5 [3] h20 f1 [4] h20 f1 [4] h1 P h1 h5 [4] h1 h5 c0 [4] 3 h2 h5 [4] h32 h5 [4] c2 [3]

c2 [3]

Now since in the ASS of P14 the element d30 [1] detects η 2 κ2 [1], which is killed by an Atiyah-Hirzebruch differential, the element d30 [1] must be killed by an Adams differential! To prove that it is v[2] that kills it, we simply rule out other possibilities. Other elements that could support a differential and kill d30 [1] are z[3], h1 P h1 h5 [3], h20 f1 [4], h32 h5 [4], c2 [3]. (∗) They all come from the 3 and 4-cells. We therefore consider the quotient map P14 → P34 and use the Mahowald square to prove that, except for one of them which supports an Adams differential—so is irrelevant for the differential we want to prove—the classes in (∗) are all permanent cycles. This then gives us a proof that d3 (v[2]) = d30 [1] in the ASS of P14 and therefore in the ASS of P1∞ , which completes the proof of the differential d3 (d0 r) = h1 d30 in the ASS of the sphere spectrum.

264

ZHOULI XU

For the five classes, it is straightforward to check that they all survive in the ASS of P34 by using the method of Mahowald square as explained in Section 2. In fact, they detect homotopy classes ηκ2 [3], η{P h1 h5 }[3], η{f1 }[3], ν{h22 h5 }[4], {f1 }[4]. For the class {f1 }[4], we have an Atiyah-Hirzebruch differential in P14 : d2 ({f1 }[4]) = η{f1 }[2]. This shows that the element h1 f1 [2] must be killed in ASS of P14 and the following Adams differential is the only possibility: d2 (c2 [3]) = h1 f1 [2]. For the other homotopy classes, it is straightforward to check that they all survive in the AHSS of P14 . Therefore, by the naturality of the AHSS, the corresponding homotopy classes must be detected by elements in the ASS of P14 , with Adams filtration at most the same as the ones in the ASS of P34 . This proves that the other 4 classes z[3], h1 P h1 h5 [3], h20 f1 [4], h32 h5 [4] are all permanent cycles. Following the “zigzag,” we could summarize the idea of the proof by the “road map”: / P1∞ / S0 P14 P34 o As a general comment on the RP ∞ -method, it is clear that the method works better when the “spheres of origin” of the source and target are in nearby dimensions. When the difference of the dimensions is large, one usually needs to run a longer “zigzag.” For example, in the proof of the Adams differential d3 (D3 ) = B3 , the “spheres of origin” of the source and target are in dimensions 22 and 6. See Appendix II of [26] for the “zigzag” process of the proof of the Adams differential d3 (D3 ) = B3 . 4. The Kahn-Priddy map and Toda brackets The Kahn-Priddy theorem implies that any homotopy class of the sphere spectrum in stem n must factor through P1n . We illustrate by examples that this observation and the computations in [28] lead us to Toda brackets in some generalized sense. We use the theory of cell diagrams to denote these Toda brackets. For background and uses of cell diagrams, see [2, 26, 29] for example. Consider the homotopy class ∈ π8 . It is detected by c0 . The pre-image under the algebraic Kahn-Priddy map is h22 [2]. Since the 1-cell in P1∞ maps to S 0 by η, we have the following cell diagram for . '&%$ !"# 8 >> >> ν 2 >> >>  !"# '&%$ 2 2

'&%$ !"# 1 >> >> η >> >>  !"# '&%$ 0 In other words, we have a Toda bracket ∈ ν 2 , 2, η in π8 .

MAHOWALD SQUARE AND ADAMS DIFFERENTIALS

265

As another example, take κ ∈ π14 . It is detected by d0 . The pre-image under the algebraic Kahn-Priddy map is c0 [6]. Since the minimal HF2 -subcomplex of P1∞ that contains the 6-cell is the 3 cell complex with cells in dimension 6, 5 and 3, and the 3-cell maps to S 0 by ν, we have the following cell diagram for κ. 7654 0123 14@ @@ @@ @@ @ '&%$ !"# 6 2

'&%$ !"# 5 η

'&%$ !"# 3 >> >> ν >> >>  !"# '&%$ 0 In other words, we have a Toda bracket κ ∈  , 2, η, ν in π14 . Let’s consider κ ∈ π20 . It is detected by g. The pre-image under the algebraic Kahn-Priddy map is h0 h23 [6]. Now note that h0 h23 does not survive in the ASS of the sphere spectrum. However, as discussed in Section 2, h0 h23 [1] survives in the mod 2 Moore spectrum and detects κ[1]. We have P56 as a suspension of the mod 2 Moore spectrum. Therefore, we have the following cell diagram for κ. 7654 0123 20@ @@ @@κ @@ @ '&%$ !"# 6 2

'&%$ !"# 5 η

'&%$ !"# 3 >> >> ν >> >>  !"# '&%$ 0 In other words, we have a Toda bracket κ ∈ κ, 2, η, ν in π20 . It is clear that these observations work better when the “sphere of origin” is in low stems. Now we consider the class θ4.5 ∈ π45 defined by Isaksen in [10]. It is detected by h34 . The pre-image under the algebraic Kahn-Priddy map is h24 [15]. The 15-cell is attached to the 6-cell by ησ. Since ησθ4 = 0, we need to add in more cells to kill this obstruction. A natural candidate is the 8-cell, since it is attached to the 6-cell by η. This raises another problem: ησ on the 6-cell would be killed by σ on the 8-cell. To avoid this problem, we add in the 7-cell as well, so σ on the 8-cell would

266

ZHOULI XU

kill 2σ on the 7-cell instead. Therefore, θ4.5 maps through the cofiber of ησ[6] for P18 . The cell diagram is given below. We can also form θ4.5 as a Toda bracket of the following maps: ⎡ 

S 44

θ4

σθ4

⎣



/ S 14 ∨ S 7

ησ[6] 0 η[6] 2

⎤

⎡

⎦

⎣

/ P16 ∨ S 7

where KP6 is the Kahn-Priddy map restricted to P16 . As a sanity check, we have * + )  )   ησ[6] , KP6 + 2, θ4 2θ4.5 = 2, θ4 σθ4 , η[6]

KP6 σ

σθ4



⎤ ⎦

/ S0

* ,

0 2

+, σ.

The second term is 2, σθ4 , 2 σ, which contains ησ 2 θ4 = 0. Note that since we know the exponent of P16 : 8 · π∗ (P16 ) = 0, we have that 16 · θ4.5 = 0, which is compatible with our knowledge. 7654 0123 45 NNN NNN θ NN4N NNN N' 0123 7654 15 σθ4

( '&%$ !"# 8 ησ

2 η

'&%$ !"# 7 '&%$ !"# 6

ν

2

'&%$ !"# 5 '&%$ !"# 4 2 η

η σ

'&%$ !"# 33 33 33 33 33 '&%$ !"# 2 33 33ν 2 33 33 '&%$ !"# 3 1 NNN NNN η 33 NNN 33 NNN 33 NN&   '&%$ !"# 0

MAHOWALD SQUARE AND ADAMS DIFFERENTIALS

267

References [1] J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. (2) 72 (1960), 20–104, DOI 10.2307/1970147. MR0141119 [2] M. G. Barratt, J. D. S. Jones, and M. E. Mahowald, Relations amongst Toda brackets and the Kervaire invariant in dimension 62, J. London Math. Soc. (2) 30 (1984), no. 3, 533–550, DOI 10.1112/jlms/s2-30.3.533. MR810962 [3] M. G. Barratt, M. E. Mahowald, and M. C. Tangora, Some differentials in the Adams spectral sequence. II, Topology 9 (1970), 309–316, DOI 10.1016/0040-9383(70)90055-8. MR0266215 [4] Robert Bruner, A new differential in the Adams spectral sequence, Topology 23 (1984), no. 3, 271–276, DOI 10.1016/0040-9383(84)90010-7. MR770563 [5] Robert Bruner. The cohomology of the mod 2 Steenrod algebra: a computer calculation. http://www.math.wayne.edu/˜rrb/papers/cohom.pdf [6] Joel M. Cohen, The decomposition of stable homotopy, Ann. of Math. (2) 87 (1968), 305–320, DOI 10.2307/1970586. MR0231377 [7] Donald M. Davis and Mark Mahowald, The image of the stable J-homomorphism, Topology 28 (1989), no. 1, 39–58, DOI 10.1016/0040-9383(89)90031-1. MR991098 [8] Donald M. Davis and Mark Mahowald, v1 - and v2 -periodicity in stable homotopy theory, Amer. J. Math. 103 (1981), no. 4, 615–659, DOI 10.2307/2374044. MR623131 [9] Bogdan Gheorghe, Guozhen Wang and Zhouli Xu. Motivic Cτ -modules and BP∗ BP comodules. In preparation. [10] Daniel C. Isaksen. Stable stems. arXiv:1407.8418. [11] Daniel C. Isaksen. Classical and motivic Adams charts. arXiv:1401.4983. [12] Daniel C. Isaksen and Zhouli Xu, Motivic stable homotopy and the stable 51 and 52 stems, Topology Appl. 190 (2015), 31–34, DOI 10.1016/j.topol.2015.04.008. MR3349503 [13] Dan Isaksen, Guozhen Wang and Zhouli Xu. More stable stems. In preparation. [14] Daniel S. Kahn and Stewart B. Priddy, The transfer and stable homotopy theory, Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 1, 103–111, DOI 10.1017/S0305004100054335. MR0464230 [15] Wˆ en Hsiung Lin, Algebraic Kahn-Priddy theorem, Pacific J. Math. 96 (1981), no. 2, 435–455. MR637982 [16] Mark Mahowald, The order of the image of the J-homomorphism, Proc. Advanced Study Inst. on Algebraic Topology (Aarhus, 1970), Mat. Inst., Aarhus Univ., Aarhus, 1970, pp. 376–384. MR0276962 [17] Mark Mahowald, The metastable homotopy of S n , Memoirs of the American Mathematical Society, No. 72, American Mathematical Society, Providence, R.I., 1967. MR0236923 [18] Mark Mahowald, A new infinite family in 2 π∗ s , Topology 16 (1977), no. 3, 249–256, DOI 10.1016/0040-9383(77)90005-2. MR0445498 [19] Mark Mahowald and Martin Tangora, Some differentials in the Adams spectral sequence, Topology 6 (1967), 349–369, DOI 10.1016/0040-9383(67)90023-7. MR0214072 [20] J. Peter May, Matric Massey products, J. Algebra 12 (1969), 533–568, DOI 10.1016/00218693(69)90027-1. MR0238929 [21] J. Peter May, THE COHOMOLOGY OF RESTRICTED LIE ALGEBRAS AND OF HOPF ALGEBRAS: APPLICATION TO THE STEENROD ALGEBRA, ProQuest LLC, Ann Arbor, MI, 1964. Thesis (Ph.D.)–Princeton University. MR2614527 [22] R. Michael F. Moss, Secondary compositions and the Adams spectral sequence, Math. Z. 115 (1970), 283–310, DOI 10.1007/BF01129978. MR0266216 [23] Douglas C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics, vol. 121, Academic Press, Inc., Orlando, FL, 1986. MR860042 [24] Martin C. Tangora, On the cohomology of the Steenrod algebra, Math. Z. 116 (1970), 18–64, DOI 10.1007/BF01110185. MR0266205 [25] Hirosi Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies, No. 49, Princeton University Press, Princeton, N.J., 1962. MR0143217 [26] Guozhen Wang and Zhouli Xu, The triviality of the 61-stem in the stable homotopy groups of spheres, Ann. of Math. (2) 186 (2017), no. 2, 501–580, DOI 10.4007/annals.2017.186.2.3. MR3702672 [27] Guozhen Wang and Zhouli Xu. The algebraic Atiyah-Hurzebruch spectral sequence of real projective spectra. arXiv:1601.02185.

268

ZHOULI XU

[28] Guozhen Wang and Zhouli Xu, Some extensions in the Adams spectral sequence and the 51-stem, Algebr. Geom. Topol. 18 (2018), no. 7, 3887–3906, DOI 10.2140/agt.2018.18.3887. MR3892234 [29] Zhouli Xu, The strong Kervaire invariant problem in dimension 62, Geom. Topol. 20 (2016), no. 3, 1611–1624, DOI 10.2140/gt.2016.20.1611. MR3523064 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Email address: [email protected]

CONM

729

ISBN 978-1-4704-4244-6

9 781470 442446 CONM/729

Homotopy Theory: Tools and Applications • Davis et al., Editors

This volume contains the proceedings of the conference Homotopy Theory: Tools and Applications, in honor of Paul Goerss’s 60th birthday, held from July 17–21, 2017, at the University of Illinois at Urbana-Champaign, Urbana, IL. The articles cover a variety of topics spanning the current research frontier of homotopy theory. This includes articles concerning both computations and the formal theory of chromatic homotopy, different aspects of equivariant homotopy theory and K-theory, as well as articles concerned with structured ring spectra, cyclotomic spectra associated to perfectoid fields, and the theory of higher homotopy operations.