This book describes an invariant, l, of oriented rational homology 3-spheres which is a generalization of work of Andrew

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- Kevin Walker

*Table of contents : Contents0 Introduction1 Topology of Representation Spaces2 Definition of λ3 Various Properties of λ4 The Dehn Surgery Formula5 Combinatorial Definition of λ6 Consequences of the Dehn Surgery FormulaA Dedekind SumsB Alexander PolynomialsBibliography*

Annals of Mathematics Studies Number 126

An Extension of Casson’s Invariant by Kevin Walker

PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY 1992

Copyright © 1992 by Princeton University Press ALL RIGHTS RESERVED

The Annals of Mathematics Studies are edited by Luis A. Caffarelli, John N. Mather, and Elias M. Stein

Princeton University Press books are printed on acid-free paper, and meet the guidelines for permanence and durabil ity of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources

Printed in the United States of America by Princeton University Press, 41 William Street Princeton, New Jersey

Library of Congress Cataloging-in-Publication Data Walker, Kevin, 1963An extension of Casson’s invariant / Kevin Walker. p. cm .— (Annals of mathematics studies; no. 126) Includes bibliographical references. ISBN 0-691-08766-0 (CL)— ISBN 0-691-02532-0 (PB) 1. Three-manifolds (Topology) 2. Invariants. I. Title. II. Title: Casson’s invariant. III. Series. QA613.W 34 1992 514.3— dc20 91-42226

Contents 0 Introduction

3

1 Topology of Representation Spaces

6

2 Definition of

27

X

3 Various Properties of A,

41

4 The Dehn Surgery Formula

81

5 Combinatorial Definition of X

95

6 Consequences of the Dehn Surgery Formula

108

A Dedekind Sums

113

B Alexander Polynomials Bibliography

122 129

v

An Extension of Casson’ s Invariant

Introduction In lectures at MSRI in 1985, Andrew Casson described an integer valued invariant A of oriented homology 3-spheres (see also [AM]). A(M) can be thought of counting the number of conjugacy classes of representations 7Ti(M) —► SU(2), in the same sense that the Lefschetz number of a map counts the number of fixed points. (Warning: According to Casson’s origi nal definition and [AM], \(M ) is half this number.) More precisely, let (W\ , W2, F) be a Ileegaard splitting of M (see (l.A)). For X any space, let R(X) denote the space of conjugacy classes of repre sentations TTi(Af) —» SU(2). We can make the identification R(M) = H(Wi) H R(W2) C R(F). R(W\) and R(W 2 ) have complementary dimensions in R(F), and, roughly speaking, A(M) is defined to be the intersection number A(A/)Hf (# (m ), W ) ) R(Wj) and R(F) are not manifolds, but singular real algebraic sets, so it might seem that there is some difficulty in defining the above intersection number. But the condition that H\(M\ Z) = 0 guarantees that after ex cising the trivial representation R(W\) DR(W 2 ) is a compact subset of the non-singular part of R(F). Thus, after an isotopy supported away from the singularities of R(F), R(W\) fl R(W 2 ) consists of a well-defined (signed) number of non-singular points, and (R(W\), R(W 2 )) is defined to be this number. It is not hard to show that this does not depend on the choice of Heegaard splitting. It is clear from the definition that A(M) = 0 if 7Ti (A/) = 1. In particular, (0.1) A(S'3) = 0. 3

4

§ 0. INTRODUCTION

Casson also showed that A has the following properties. (0.2) Let K be a knot in an integral homology sphere and let ATi/n denote 1/n Dehn surgery on K. Let A#(*) be the (suitably normalized) Alexander polynomial of K. Then \ ( K 1/n) = X(Kl/0) + n ^ A K(l). (0.3) A(—M) = —A(M), where —M denotes M with the opposite orienta tion. (0.4) A(Mi #M 2) = A(Mi) + A(M2), where # denotes connected sum. (0.5) 4A(M) = n(M ) (mod 16), where /i(M) denotes the signature of a spin 4-manifold bounded by M. It is not hard to show that (0.1) and (0.2) uniquely determine A. Also, (0.3), (0.4) and (0.5) follow easily from (0.1) and (0.2). The proof of (0.2) involves clever exploitation of the isotopy invariance of (R(Wi), R(W 2 )). For a more detailed overview of A for integral homology spheres, see the introduction of [AM]. This paper contains generalizations of the above results to the case where M is a rational homology sphere (i.e. H*(M) Q) 2 * i/*(S 3 ;Q), or |#i(M ; Z)| < oo). In this case, R(Wi)C\R(W 2 ) contains nontrivial singular points, so finding an isotopy invariant definition of (R(Wi), R(W 2 )) is more difficult. It can, nevertheless, be done, and one can use the isotopy invari ance to prove a generalization (0.2) (see (4.2)), and hence generalizations of (0.3), (0.4) and (0.5) (see (6.5)). For a more detailed introduction, the reader is encouraged to read (l.A) and the beginning of ( 2 .A). Just as in the integral homology sphere case, the generalized Dehn surgery formula and the fact that A(S3) = 0 uniquely determine A for all rational homology spheres. In this case, however, it is not very hard to prove that the Dehn surgery formula does not overdetermine A. In other words, one can use the Dehn surgery formula to define A and avoid SU(2) representations altogether. Section 1 contains results on the topology (and symplectic geometry) of

§ 0. INTRODUCTION

5

certain representation spaces, as well as other miscellaneous background material. Most of the results are summarized in (l.A), and the nonmethodical reader may wish to read only this subsection, refering back to the rest as necessary. Section 2 contains the definition of A (i.e. of (R(Wi), ^(W^)))- Sections 3 and 4 contain the proof of the Dehn surgery formula. Section 3 contains the parts of the proof involving representation spaces, while Section 4 contains the parts involving the manipulation of surgery diagrams. Some readers may wish to start with Section 4 and re fer back to Section 3 as necessary. In Section 5 we use the Dehn surgery formula to give an independent and elementary proof of the existence of A. This section is independent of sections 1 through 4, and some readers may wish to act accordingly. In Section 6 we use the Dehn surgery formula to prove various things about A, including its relation to the //-invariant. Sections A and B are appendices containing background material needed for the statement and proof of the Dehn surgery formula. This work has previously been announced in [Wl]. Certain results found in this paper were obtained independently by S. Boyer and D. Lines [BL] (see (5.1) and (6.5)). A different generalization of Casson’s invariant has been described by S. Boyer and A. Nicas [BN]. This research received generous support from the National Science Foun dation, the Sloan Foundation and the Mathematical Sciences Research In stitute. It is a pleasure to acknowledge helpful conversations and correspondence with Peter Braam, Dan Freed, William Goldman, Lucien Guillou, Michael Hirsch, Morris Hirsch, Steve Kerckhoff, Gordana Matic, Paul Melvin and Tom Mrowka. Christine Lescop gave an earlier version of this paper a very thorough reading and spotted numerous errors, at least one of which was very non-trivial. Very special thanks are due to Andrew Casson, whose excellent suggestions improved this paper in many ways, and to Rob Kirby, who provided good advice on all sorts of topics. Finally, I would like to thank Jose Montesinos for suggesting that I work on this problem.

T opology of Representation Spaces

A. Summary of Results and Notation. Let M be an oriented rational homology 3-sphere (QHS) with a genus g Heegaard splitting (U'i. U7. F). (That is, M — U’i U!V2 and Wi fl W 2 = dWi = dWi = F, a surface of genus g.) Let F* be F minus a disk. We have the following diagram of fundamental groups -

*i(F)

/

\

\

/

M M ).

*i(W2) Note that all maps are surjections. Applying the functor Hom( • ,517(2)), this diagram becomes R*

Q\

/ \

\ /

Q jnQ 'a

Q\

(i.e. Ri =f Hom(xi(F),5£f(2)), Q] =f Hom(7n(Pyj ), 517(2)), etc.). Note that all maps are injections and that Van Kampen’s theorem implies that Hom(7Ti(M),S{7 (2)) = Q\ C\Q\ • SU( 2 ) acts naturally on these spaces via 6

§ 1. TOPOLOGY OF REPRESENTATION SPACES

7

the adjoint action. Taking quotients, we get R*

R

Qi

/ \

\

Q 1 OQ 2

/ Q2

(i.e. R = Hom(Tn(F),5t/(2))/5f7(2), etc.). In the rest of this paper the superscript jj will be used, without comment, to denote the “inverse quotient by the adjoint action of S{7 (2 )”, and viceversa. That is, if X been defined as Y/S(7(2), then will be used to denote Y . We will see below that R has two singular strata: S, consisting of (equiv alence classes of) representations with abelian image, and P, consisting of representations into Z2 , the center of SU(2). Define def Tj R~ QJ STf

def def def def

Q jH S R \S Qj

\

Tj

S\P Tj\P .

The Zariski tangent space of R* at a representation p can be identified with the cocycle space Z 1 ( 7Ti(F); Adp), and the tangent space of the orbit through p corresponds to the coboundaries Bl (iri(F); Ad p). Thus we define the “Zariski tangent space” of R at [p] to be H l(wi(F); Adp). Let B : su(2) x su(2) —►R be an Ad-invariant inner product. B induces the cup product u B : ffV ifF ); Adp) x H l {xi(Fy,Mp) -> ffx(in(F);R) “ R. It is a result of Goldman that u>b gives R a symplectic structure. Q\ and Q 2 turn out to be lagrangian with respect to We will be particularly interested in the normal bundle of S~ in R. Define V def — Zariski normal bundle of S~ in R def m = Zariski normal bundle of T'j~ in Qj £ def actual (singular) normal bundle of S~ in R 0j = actual (singular) normal bundle of Tj~ in Qj

8

§ 1. TOPOLOGY OF REPRESENTATION SPACES

It is another result of Goldman that for p = [p\ E 5 ” , the fiber £p is diffeomorphic to a quadratic cone in vp modulo stab (p) ^ S 1. For p E Tj~, 6 j iP is diffeomorphic to rjjiP modulo stab(p). Let S q be a fixed, oriented maximal torus of SU(2) and let Z2 C S q denote the center of SU(2). Let S = E o m i M F ^ S l ) ~ (5q)2j 5Hom(7r1 (F ), 5 ^)\H om ( 7r1 (F),Z 2) - (55) 2n ( Z 2)2s 7} 1§f H o m O ^ ) ^ ) “ (StY T ~ = Hom(x 1 ( ^ i ), 5 ^)\H om ( 7r1 (W ,),Z2) - (5 01 ) '\( Z 2)'. 5 “ [Tj ] is a double cover of P}“]. Let i/, rjj, ^ and continue to denote their lifts to 5 ” or 7) , as the case may be. Over 5 “ , v is a symplectic vector bundle, and t]j is an oriented lagrangian subbundle defined over Tj . Picking a metric of F induces an hermetian structure on T R compatible with a T h i s converts v into a hermetian vector bundle and rjj into a totally real subbundle. Let det1 (i/) denote the unit vectors in the determinant line bundle det(i/) of v. rjj induces a section det 1 (r}j) of det1 (i/) over Tj . det(iz) [det(77; )] extends smoothly and canonically over S [Tj]. Now we come to a crucial point. Namely, Ci(det(i/)) = is repre sented by a multiple u> of ujb restricted to 5“ and lifted to S. Furthermore, Tj is lagrangian with respect to u>. The above results are proved (or referenced) in sections B and C. Sec tion D uses results of Newstead to prove that certain maps of representation spaces act trivially on rational cohomology. Section E is concerned with a class of isotopies of R appropriate to the definition of A. Section F estab lishes orientation conventions. Section G establishes conventions for Dehn twists and Dehn surgeries.

B. Fundamental Results of Goldman. The material in this section is treated in more detail in [Gl] and [G2]. Let 7r be a finitely presented group and G be a Lie group. Let p : n —►G be a representation and let pt : n —►G be a differentiable 1 -parameter

§ 1. TOPOLOGY OF REPRESENTATION SPACES

9

family of representations such that po = p. Writing pt(x) = exp(iu(a:) + 0 (t 2 ))p(x) (for x G 7T and t near 0) and differentiating the homomorphism condition (1-1) Pt(xy) = pt(x)pt(y), we find that (1.2) u(xy) = u(x) 4* Ad p(x) u(y). In other words, u : ir —* g is a 1-cocycle of 7r with coefficients in the irmodule gAdp- Conversely, solutions of (1.2) lead to maps pt : tt —* G which satisfy (1.1) to first order in t. Thus the Zariski tangent space of Hom(7r, G) at p can be identified with Z l {ir\gxdp). We now compute the tangent space of the Ad-orbit containing p. Let gt be a path in G with go = 1. Let Pt (*) = 9 Tl p ( x )9t-

If gt = exip(tuo + 0(^2)), then the cocycle corresponding to pt is u(x) = Adp(x)uo — uo. In other words, u is the coboundary 6 uq. Thus the tangent space of the Ad-orbit through p is 5 1 (7r;gAdp)The above results suggest that we define the “Zariski tangent space” of Hom(7r, G) at [p] to be i/H^gAdp)- (We will omit the scare quotes from now on.) We now specialize to the case 7r = 7Ti (F) and G is a Lie group which affords an Ad-invariant, symmetric, non-degenerate bilinear form B on its Lie algebra (e.g. a reductive group with the Cartan-Killing form). If X C G, let Z(X) denote the centralizer of X in G. Let Z(p) = Z(p(n)). (1.3) Proposition (G oldm an). The dimension of Z 1 (n;gAdp) is (2

g —1 ) dimG + dim Z(p).

The dimension o /i? 1 (7r;gAdp) is (2g — 2) dimG + 2 dim Z(p).

□

10

§1. TOPOLOGY OF REPRESENTATION SPACES

Note that dimZ 1 (7r;gAdp) is minimal, and hence p is a nonsingular point of Hom(7r, G), if and only if dim(Z(p)/Z(G)) = 0. We denote the set of all such points as Hom(7r,G)"". If p E Hom(7r,G) is a singular point (i.e. if dim(Z(p)/Z(G)) > 0) it is still a nonsingular point of the subvariety Hom(7r, Z(Z(p)))~. Note also that for all a E Hom(7r, Z(Z(p)))~, stab(cr(7r)) = Z(a) = Z(p). Therefore all points of Hom(7T, Z(Z(p)))~ have the same orbit type. In fact, (1.4) Proposition (G oldm an). The sets of the form Hom(7t,Z (Z (X )))-/N g (Z(Z(X ) ) ) 9 where N q denotes the normalizer in G, give a stratification o/Hom(7r, G)/G.

□

Let |p] E Hom(7r,G )/G and [u] E # SAdp)- We wish to find nec essary and sufficient conditions for [u] to be tangent to a path [pt] in Hom(7r, G)/G. Writing P t ( x ) = exp(«u(.r) 4- t 2 v(x) + 0(t 3 ))p(x), plugging this into ( 1 .1 ), and expanding to second order in t, we find

v(x) -v (x y ) -f Ad p(x)v(y) = ^[u(x), Adp(x)u(y)]. The left hand side is just the coboundary of the 1-cochain v. The right hand side is the cup product (on the cocycle level) of u with itself, using the Lie bracket [ • , • ] : g x g —►g as coefficient pairing. Thus a necessary condition for [u] to be tangent to a path in IIom(7r, G)/G is [[«],[«]] = 0 e # 2(7r;gAdp)It turns out that for surface groups this condition is also sufficient: (1.5) Theorem (G oldm an). Let [p] E IIom(7r, G)/G and a E H 1 (7r; gAd/?)Then a is tangent to a path in IIom(7r, G)/G if and only if [a, a] = 0. □ Also, (1.6) Theorem (G oldm an and M illson, [GM]). A point [p] ElIom( 7r,G)/G has a neighborhood diffeomorphic to {a G # 1 (*■;gAdp) | [a, a] = 0}/stab (p).

□

§ 1. TOPOLOGY OF REPRESENTATION SPACES

11

Recall that we have an Ad-invariant, symmetric, nondegenerate bilinear form B : g x g —>R. This induces a cup product ujB : H l (v, gAdp) x tfH ’ngAdp) -f H 2 (w, R) “ R. Regarding H 1 (w]gAdp) as the Zariski tangent space to Hom(7r, G)/G at [p], ljb defines a 2-tensor on Hom(7r,G)/G. (1.7) T heorem (G oldm an), lob is a closed, nondegenerate exterior 2form (i.e. a symplectic structure) on IIom(7r,G)/G. D ljb is compatible with the stratification of Hom(7r, G)/G in the following sense. Let [p] G Hom(7r,G)/G and let {pj} —►p. Then there is a natural inclusion i : lim # ^jngAdp,-) ->• H'fagxdp), and limu>[Pi] = i*«w . (Here ux denotes lob restricted to Txllom(ir,G)/G.) We now apply the above results to the case G = SU(2). Let Sq be a fixed maximal torus of SU(2 ). Up to conjugacy, SU(2 ) has three subgroups of the form Z(Z(X)): SU(2), Si and Z(SU(2)) = Z2. Thus, by (1.4), R (= IIom(7r, 5 J7 ( 2 ))/ 5 f/( 2 )) has two singular strata: S ^ R o m (* ,S h )/l2 '* (S tfi/Z 2 (here Z2 = JV(Sg)/Z(Sg)) and P = f Hom(x,Z 2) S Z |'. This stratification is compatible with Qj. That is, Qj has singular strata Tj = Q j H S £ ( S l y / Z 2 and Qj n P “ Zf. Let p = [p] £ Q j . The inclusion i : F Wj induces an injection i* : H 1(7r1( W j ) ; g Adp) -* P^TngAdp) which corresponds to the inclusion TpQj —►TPR. Since H 2 (n\(Wj)\R) = 0, ub is zero on ^ (ir ^ W j) ; gAdp). Furthermore, dimH 1 (7Ti(Wj)] gAdp) = \ dim if 1 (7r;gAdp)- Hence Qj is lagrangian with respect to lobIf p G P , then Ad p is trivial, and hence H l{^{ X

) ; g Adp) =

H 1(T1( X ) - , R ) ® g

12

§ 1. TOPOLOGY OF REPRESENTATION SPACES

for X = F or Wj. Since M is a QIIS, JT1 (tti(VTi);R) and H 1 (iri(W2 );R) are transverse in /f 1 ( 7Ti(F); R). Therefore Q\ and Q 2 are transverse at P.

C. The Normal Bundle of S. (1.8) The Topology of the N orm al Bundle. Let p = [p] € 5~. Let h be the Lie algebra of S q and h -1 be the orthogonal complement of h (with respect to B ). Then the 7r-module gAdp decomposes as liAdp ® h id p, and hence gAdp) = The space H x(ir; h ^ dp) corresponds to the fiber of the Zariski normal bundle of S'” in iZ at p. Call this bundle v. Similarly, if p € Tj~ also, then H 1 (/^ i(Wj)]h'^dp) is the fiber of the Zariski normal bundle of T~ in Qj at p. Call this bundle r)j. Let £ and Oj denote the actual normal bundles of 5 “ and Tj~. By (1.5), we have £ = {x € v | [a:,x] = 0 }/Sl Oj = Vj/So(Here should be thought of as stab (Sq) = stab (p(7r)).) Let £ denote the unit vectors in £ [Oj]. It will be convenient to put a hermetian structure on v compatible with its symplectic structure. This can be done as follows. Let h£d denote the flat h^-bundle over F with holonomy Ad p. (The ambiguity in notation is intentional.) Let H\R(F\liAdp) Rham cohomology group. Since Wi(F) = 0for i > 2, we have H\R(F\h^dp) = ^ 1 ( 7r5^1Adp)> an^ we identify these two spaces from now on. Choose a metric on F .Note that H q(F]hj^dp) can also be identified with the space of harmonic g-forms with coefficients in h i dp. Choose B to be positive definite (e.g. the negative of the Cartan-Killing form). The metric on F, together with B, induces the Hodge star operator * : H 'i * ;h^dp) — and the Hodge metric S

q

[ Qj ]

§ 1. TOPOLOGY OF REPRESENTATION SPACES where

(a i P)

13

J B(a, */?).

It is easy to see that * and ( • , •) give a hermetian structure compatible with us- That is, = -(