Geometry, Topology, and Dynamics

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Selected Title s i n Thi s Serie s Frangoi s Lalonde , Editor , Geometry , topology , an d dynamics , 1 99 8 J o h n Harna d an d A l e x K a s m a n , Editors , Th e bispectra l problem , 1 99 8 Miche l C . Delfour , Editor , Boundaries , interfaces , an d transitions , 1 99 8 P e t e r C . Greiner , V i c t o r Ivrii , Lui s A . Seco , an d Catherin e S u l e m , Editors , Partial differentia l equation s an d thei r applications , 1 99 7 11 Lu c V i n e t , Editor , Advance s i n mathematica l sciences : CRM' s 2 5 years , 1 99 7 10 D o n a l d E . K n u t h , Stabl e marriag e an d it s relatio n t o othe r combinatoria l problems : A n introduction t o th e mathematica l analysi s o f algorithms , 1 99 7 9 D . Levi , L . V i n e t , an d P . W i n t e r n i t z , Editors , Symmetrie s an d integrabilit y o f difference equations , 1 99 6 8 J . Feldman , R . Froese , an d L . M . R o s e n , Editors , Mathematica l quantu m theor y II : Schrodinger operators , 1 99 5 7 J . Feldman , R . Froese , an d L . M . R o s e n , Editors , Mathematica l quantu m theor y I : Field theor y an d many-bod y theory , 1 99 4 6 G u i d o Mislin , Editor , Th e Hilto n Symposiu m 1 993 : Topic s i n topolog y an d grou p theory, 1 99 4 5 D . A . D a w s o n , Editor , Measure-value d processes , stochasti c partia l differentia l equations, an d interactin g systems , 1 99 4 4 H e r s h y Kisilevsk y an d M . R a m M u r t y , Editors , Ellipti c curve s an d relate d topics , 1994 « 3 R e m i Vaillancour t an d Andre i L . Smirnov , Editors , Asymptoti c method s i n mechanics, 1 99 3 2 Phili p D . L o e w e n , Optima l contro l vi a nonsmoot h analysis , 1 99 3 1 M . R a m M u r t y , Editor , Thet a functions : Fro m th e classica l t o th e modern , 1 99 3 15 14 13 12

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https://doi.org/10.1090/crmp/015

Volume 1 5

CRM PROCEEDINGS & LECTURE NOTES Centre d e Recherehe s Mathematique s Universite d e Montrea l

Geometry, Topology , and Dynamic s Frangois Lalond e Editor

The Centr e d e Recherehe s Mathematique s (CRM ) o f th e Universite d e Montrea l wa s create d i n 1 96 8 t o promot e research i n pur e an d applie d mathematic s an d relate d disciplines. Amon g it s activitie s ar e specia l them e years , summer schools , workshops , postdoctora l programs , an d publishing. Th e CR M i s supporte d b y th e Universit e d e Montreal, th e Provinc e o f Quebe c (FCAR) , an d th e Natural Science s an d Engineerin g Researc h Counci l o f Canada. I t i s affiliate d wit h th e Institu t de s Science s Mathematiques (ISM ) o f Montreal , whos e constituen t members ar e Concordi a University , McGil l University , th e Universite d e Montreal , th e Universit e d u Quebe c a Montreal, an d th e Ecol e Polytechnique . Th e CR M ma y be reache d o n th e We b a t www.crm.umontreal.ca .

-\Y% America n Mathematical Societ y fe Providence , Rhode Island US A

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T h e p r o d u c t i o n o f thi s volum e wa s s u p p o r t e d i n p a r t b y t h e Fond s p o u r l a F o r m a t i o n de Chercheur s e t l'Aid e a l a Recherch e (Fond s F C A R ) a n d t h e N a t u r a l Science s a n d Engineering Researc h Counci l o f C a n a d a ( N S E R C ) . 1991 Mathematics Subject

Classification.

Primar

y 03C1 5 , 58Dxx , 58Fxx .

Library o f Congres s Cataloging-in-Publicatio n D a t a Geometry, topology , an d dynamic s / Frangoi s Lalonde , editor . p. cm . — (CR M proceeding s & lectur e notes , ISS N 1 065-858 0 ; v. 1 5 ) Proceedings o f th e Worksho p o n Geometry , Topology , an d Dynamics , hel d a t th e CRM , Universite d e Montreal , Jun e 26-30 , 1 995 . Includes bibliographica l references . ISBN 0-821 8-0877- X (alk . paper ) 1. Geometry , Differential—Congresses . 2 . Differentia l topology—Congresses . 3 . Differen t i a t e dynamica l systems—Congresses . I . Lalonde, Frangois . II . Workshop o n Geometry , Topol ogy, an d Dynamic s (1 99 5 : Centr e d e recherche s mathematiques , Universit e d e Montreal ) III. Series . QA641.G4471 99 8 551.48—dc21 98- 342 8 CIP

C o p y i n g an d reprinting . Materia l i n this boo k ma y b e reproduced b y any mean s fo r educationa l and scientifi c purpose s withou t fe e o r permissio n wit h th e exceptio n o f reproductio n b y service s that collec t fee s fo r deliver y o f documents an d provide d tha t th e customar y acknowledgmen t o f th e source i s given. Thi s consen t doe s no t exten d t o othe r kind s o f copying fo r genera l distribution , fo r advertising o r promotiona l purposes , o r fo r resale . Request s fo r permissio n fo r commercia l us e o f material shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematica l Society , P. O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o b e mad e b y e-mai l t o [email protected]. Excluded fro m thes e provision s i s materia l i n article s fo r whic h th e autho r hold s copyright . I n such cases , request s fo r permissio n t o us e o r reprin t shoul d b e addresse d directl y t o th e author(s) . (Copyright ownershi p i s indicate d i n th e notic e i n th e lowe r right-han d corne r o f th e firs t pag e o f each article. ) © 1 99 8 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s pxr.p.pt thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America . @ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . This volum e wa s typese t usin g AJ^S-TEK an d Aj^[S-WT^>i, the America n Mathematica l Society' s T E ^ macr o system , and submitte d t o th e America n Mathematica l Societ y i n camer a read y form b y th e Centr e d e Recherche s Mathematiques . Visit th e AM S hom e pag e a t URL : http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 0

3 02 01 00 99 9 8

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Contents

Preface vi

i

Isomorphisms Betwee n Classica l Diffeomorphis 1 m Group s Augustin Banyaga Classification o f Topologicall y Trivia 1 l Legendria n Knot s Yakov Eliashberg and Maia Fraser

7

Contact Structure s o n 7-Manifold s 5 Hansjorg Geiges and Charles B. Thomas

3

On th e Flu x Conjecture s 6 Francois Lalonde, Dusa McDuff, and Leonid Polterovich

9

About th e Bubblin g Of f Phenomeno n i n the Limi t o f a Sequence of J-Curves 8 Veronique Lizan

7

Symplectic Resolutio n o f Isolate d Algebrai 1 c Singularitie s 0 1 John D. McCarthy and Jon G. Wolf son Generating Functions versus Action Functional—Stable Mors e Theory versu s Floer Theor y 0 Darko Milinkovic and Yong-Geun Oh Scalar Curvatur e Rigidit y o f Certai1 n Symmetri c Space s 2 Maung Min-Oo Bi-Invariant Metric s fo r Symplecti c Twis t Mapping s o n T*T n an d a n Appli 1 cation i n Aubry-Mathe r Theor y 3 Karl Friedrich Siburg

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7 7

7

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Preface The Worksho p o n Geometry , Topolog y an d Dynamic s wa s hel d a t th e CRM , Universite d e Montreal, fro m Jun e 2 6 to Jun e 30 , 1 995 . I t too k plac e at a n interest ing momen t wit h respec t t o symplecti c developments . Durin g th e precedin g year , Seiberg an d Witte n ha d just introduce d thei r famou s Gaug e equations , fro m whic h Taubes extracte d ne w invariants , which , moreover , wer e show n t o b e equivalen t i n some sens e t o a particula r for m o f the Gromo v invariant s fo r symplecti c manifold s in rea l dimensio n 4 . This spectacula r developmen t i s a crucia l piec e o f the J-holomorphi c approac h in symplectic geometry . Indeed , Gromov' s theor y i s a deformation theory : startin g with an y know n (sa y Kahler ) model , on e ca n us e Gromov' s theor y t o prov e th e persistence o f som e familie s o f pseudo-holomorphi c curve s an d deduc e fro m thi s new informatio n t o bette r understan d symplecti c manifold s whic h ar e deformatio n equivalent t o Kahle r manifolds . Wit h Taubes ' equivalenc e theorem , th e existence side of the theor y become s available , an d on e can a priori establis h th e existenc e of J-holomorphic curve s o n manifold s usin g only , say , th e existenc e o f a metri c wit h positive scalar curvature . A t th e sam e time, contac t geometr y wa s rapidly develop ing. Usin g bot h holomorphi c argument s i n symplectization s o f contac t manifolds , ad ho c topologica l arguments , o r gaug e theoreti c methods , severa l result s o n 3 dimensional contac t manifold s wer e obtained , an d ne w surprisin g fact s abou t th e Bennequin-Thurston invarian t wer e derived . Furthermore, Hofer' s geometry , th e intrinsi c geometr y o n th e grou p o f Hamil tonian diffeomorphism s o f a symplecti c manifold , ha d jus t bee n developed . Ther e is a fascinatin g relatio n betwee n th e Hofe r norm , th e pseudo-holomorphi c curve s approach, an d th e if-are a recentl y introduce d b y Gromov . Finally, lon g lasting conjectures concernin g the fact tha t th e group of Hamilton ian diffeomorphism s b e C 1 (o r eve n C°) close d i n th e grou p o f al l diffeomorphism s of a symplectic manifold pla y a fundamental rol e in our understanding o f symplecti c invariants, namel y Floe r (o r Floer-Novikov ) homologies . These development s an d conjecture s constitut e th e genera l contex t i n whic h the worksho p too k place . Mos t o f th e talk s wer e origina l contribution s t o som e of th e subject s mentione d above . Th e followin g peopl e gav e one-hou r lectures : A. Banyaga , M . Bialy , J . Bland , Y . Eliashberg , E . Giroux , V . Ginzburg , H . Hofer , D. Kotschick , F . Laudenbach , S . Makar-Limanov , D . McDuff , M . Min-Oo , L . Moatty, Y . G . Oh , L . Polterovich , D . Salamon , K . F . Siburg , J . C . Sikorav , C . B . Thomas, L . Traynor , I . Ustilovsk y an d J . Wolfson . I wis h t o than k NSER C (Canada ) an d FCA R (Quebec ) fo r partia l fundin g o f this event . I a m mos t o f al l gratefu l t o th e CR M fo r a larg e par t o f th e fundin g and fo r it s efficac y an d enthusias m i n organizin g th e workshop . Specia l thank s

Vll

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Vlll

PREFACE

to Lu c Vinet , Loui s Pelletie r an d Jose e Laferrier e fo r th e plannin g an d organisa tion, an d t o Marti n Goldstei n an d Andr e Montpeti t fo r th e typesettin g an d thei r invaluable hel p i n editin g thes e proceedings . Frangois Lalond e Montreal, Februar y 1 997 .

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https://doi.org/10.1090/crmp/015/01 Centre d e Recherche s Mathematique s CRM Proceeding s an d Lectur e Note s Volume 1 5 , 1 99 8

Isomorphisms Betwee n Classica l Diffeomorphis m Group s Augustin Banyag a ABSTRACT. W e revie w theorem s assertin g tha t classica l structure s ar e deter mined b y thei r automorphis m groups : a contributio n t o th e Erlange r Pro gramm o f Klein . W e sho w tha t th e method s use d t o prov e thes e result s yiel d analogous result s fo r measure-preservin g homeomorphisms .

1. Statemen t o f th e mai n result s In hi s "Erlange r Programm " (1 872 ) [1 6] , Klein define d a geometr y o f a se t X as th e stud y o f thos e propertie s o f "figures " (subset s o f X) whic h remai n invarian t under a grou p G(X) o f transformations o f X. Groups of transformations had been used in geometry for many years, but Klein's originality consisted in reversing the roles, in making the group the primary object of interest and letting it operate on various geometries, looking for invariants [1 5 , p. 212]. Following the spirit o f the Erlanger Programme , geometri c structures, includin g topological structures , differentiabl e structures , unimodula r structures , symplec tic structures , contac t structures , foliation s etc . should b e studied b y analyzin g th e transformation group s preservin g thes e structures . Th e automorphis m grou p o f a topological se t i s the grou p o f homeomorphisms; th e automorphis m grou p o f a dif ferentiable structur e o n a differentiable manifol d M i s the grou p o f diffeomorphis m of M. Th e automorphis m grou p o f an unimodula r structur e i s the grou p o f volum e preserving diffeomorphisms . Th e automorphism grou p of a symplectic structure , o r a contac t structur e ar e th e grou p o f symplecti c o r contac t diffeomorphisms . Fro m this poin t o f view, i t i s important t o kno w i f these automorphis m group s determin e the correspondin g structure . The cas e o f topologica l spac e ha s bee n studie d i n detai l b y Gerstenhabe r [1 4] , Fine an d Schweiger t [1 2] , Wechsle r [23 ] an d Whittake r [24] : i t i s show n tha t fo r certain topologica l spaces , includin g compac t manifold s wit h o r withou t boundar y a grou p isomorphis m betwee n thei r homeomorphis m group s i s induced b y a home omorphism betwee n thes e spaces . Mor e precisely , i f Xi, i = 1 , 2 , ar e suc h space s and i f (p: H{X\)— > H(X2) i s a grou p isomorphis m betwee n th e homeomorphis m 1991 Mathematics Subject Classification. 53C1 2 , 53C1 5 . Supported i n par t b y NS F gran t DM S 94-031 96 . This i s th e final versio n o f th e paper . ©1998 America n Mathematica l Societ y 1 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

2

A. BANYAG A

groups H(Xi) o f Xi, the n ther e exist s a homeomorphis m w: X\ — > X2 suc h tha t ^ ( / ) = n ; / i £ ; - 1 f o r a l l / G f l r ( X1 ) . The goa l o f thi s pape r i s t o surve y analogu e result s fo r th e so-calle d classica l structures: whic h compris e the differentiable structures , th e unimodular structures , the symplecti c structures , an d th e contac t structures . W e sho w tha t th e method s used exten d t o prov e analogu e result s fo r measur e preservin g homeomorphisms . Recall that a volume form o n a n-dimensional manifol d M i s an n-form w which is everywher e no n zero , tha t a symplecti c for m o n a 2m-dimensiona l manifol d i s a a close d 2-for m ft suc h tha t fl n = Q,AfLA''-AQ ( m times ) i s a volum e form , an d that a contac t for m o n a n a 2r a + 1 -dimensiona l manifol d i s a 1 -for m a suc h tha t a A (da) 1 71 i s a volum e form . Here ar e th e mai n results : THEOREM 1 . 1 (Filipkiewic z [1 1 ]) . Ifip: Diff°°(Mi ) - > Diff°°(Mi ) is a group isomorphism between the groups Diff°°(Mi ) of C°° diffeomorphisms of connected smooth manifolds Mi, then there exists a C°° diffeomorphism w: Mi — > M 2 such that Diff^ 0 (M 2 ,a 2 )

then there exists a contact diffeomorphism w: (M , a i)— » (M 2 , a2 ) such that tp(f) = wfw-1 for all f G Diff f ( M i , a i ) . T H E O R E M 1 .5 . Let (Mi,Qi) be two connected symplectic manifolds of dimension 2n > 2 satisfying the following conditions: either Mi are compact or their symplectic pairings are trivial, then any isomorphism between their symplectic diffeomorphisms is induced by a conformal symplectic diffeorphism between (Mi,fli). More precisely, if there exists an isomorphism Diff^ 2 (M 2 ) ; then there exists a diffeomorphism w: Mi — > M 2 such that u>*fi 2 = Af^ i for some constant X and such that ip(f) = wfw -1 for all f G Diffo1 (Mi) . T H E O R E M 1 .6 . Let (M^i^ ) be two n dimensional manifolds carrying volumeforms uJi, i = 1 , 2. If n > 2 and if there is an isomorphism (p: Diff Wl (Mi)— > DifFa;2(M2); then there exists a diffeomorphism w : Mi —> M 2 such thatw*u)2 = XLUI for some constant A and such that (p(f) = wfw~ 1 for all f G DiS UJl(Mi).

The topologica l versio n o f thi s resul t i s the followin g THEOREM 1 .7 . Let Mi, fa) be two compact connected oriented smooth manifolds of dimension n > 3 equipped with Radon measures fa such that no point has positive measure and such that non empty open sets have strictly positive measure. If there exists a group isomorphism (p: H^(Mi)—- > iJM 2 (M 2 ) ; where H^Mi) is the group of measure preserving homeomorphisms of (Mi, fa), then there exists a homeomorphism w : Mi— > M 2 such that ip(f) — wfw'1 for all f G Hfll(Mi) and such that w*fa = Xfa for some constant A .

The proo f o f Theorem s 1 .4 , 1 .5 , an d 1 . 6 us e als o thei r infinitesima l versions . The differentiabl e cas e wa s prove d b y Pursell-Shank s [1 9] . Omor i generalize d th e result t o contact , symplecti c an d divergenc e fre e vecto r field s [1 8] : T H E O R E M 1 . 8 (Pursell-Shanks) . Suppose there exists a Lie algebra isomorphism p : XM X ~* %M 2 , between the Lie algebras XM± of smooth vector fields on two smooth manifolds Mi, then there exists a smooth diffeomorphism w: Mi — • M2 such that p(£) = w*£ , V£ G XM1 • T H E O R E M 1 . 9 (Omori) . Let (Mi, ai),i — 1 ,2 be two smooth manifolds equipped with a volume form, a symplectic form, or a contact form, we commonly denote ai. Let C ai(Mi) be Lie algebras of contact, or symplectic, or divergence free vector fields (in case ai are contact, symplectic, or volume forms). Suppose there exists a Lie algebra isomorphism p: C ai(Mi)— * £^2 (M 2 ) ; then there exists a smooth diffeomorphism w: Mi — * M2 such that p(£) = w*£, V £ G XM 1 } ^nd such that w*a2 = Xai for some function X, which is a constant if ai are symplectic or volume forms.

The proo f w e outline below fo r Theore m 1 .2 , relie s o n th e followin g theore m o f Montgomery-Zippin [1 7] : T H E O R E M 1 .1 0 (Montgomery-Zippin) . Let $ be a homomorphism from a Lie group G into the group H(M) of homeomorphisms of a manifold of class C k, Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

4

A. BANYAG A

1 < k < oo , o;. Suppose that for each g G G the map x — i > $(g)(x) is a C k diffeomorphism of M, then the map (g,x) •— > $(g)(x) is a C k in both variables ( M 2 mus t b e a homeomorphism [20 , 1 1 ] . Let M b e a smoot h manifol d an d le t A denot e th e clas s o f fixed subset s o f elements o f Diff°°(M) , i.e .

^ = {Fix(/)|/GDiflf°°(M) } and Fix(/ ) — {x ^ M \ f{x) — x}. Le t B b e th e clas s o f complement s o f element s of A, i.e . B = {B = M - A, A G A}. B y definition , i f B G S, the n B i s a n ope n set whic h i s th e interio r o f th e suppor t o f som e diffeomorphism . Fo r an y x G M belonging t o a n ope n se t U i n U, ther e exist s a diffeomorphis m whos e suppor t contains x an d i s containe d i n U. Henc e ther e exist s B G B suc h tha t x G B an d B C U. Therefor e B i s a basi s fo r th e topolog y o f M . W e have : Fix(whw-1) =

w(Fix(h)), V/

i G Diff°°(M 1 )

1

F i x f a T ^ ) = ^ ( F i x ^ ) ) , V ^ G Diff°°(M 2) l

Both w an d w~ ar e thu s continuou s sinc e the y ma p basi c ope n set s t o basi c ope n sets. Thi s establishe s tha t w i s a homeomorphism . To prov e tha t th e continuou s map s w an d w~ l ar e C°° , i t i s enoug h t o sho w that f ow e C°°(Mi) , V / G C°°(M 2 ) an d similarl y tha t g o w~ l G C°°(M 2 ), Vg G C°°(Mi). Her e we denote by C°°(P) th e space of smooth rea l valued function s on a smoot h manifol d P. Let X b e an y C° ° vecto r field o n M\ wit h compac t suppor t an d h t th e 1 parameter grou p o f diffeomorphism s i t generates . Fo r eac h t, th e map : Ht = whtw~l i s a C° ° diffeomorphis m o f M 2 b y hypothesis , an d th e map : W : M x M 2 ^ M 2 : (t,x) ^ H t(x) ^wh

1 tw~

{x)

is continuous . Moreover , HQ — identity an d H t+S = H t o Hs. Therefor e 7i i s a continuous actio n o f the Li e group E b y smooth diffeomorphisms . B y Theorem 1 .1 0 of Montgomery-Zippin , th e ma p i s smoot h i n bot h variable s t an d x. Therefore , the 1 -paramete r grou p o f diffeomorphism s H t ha s a n infinitesima l generato r X w defined by : ^•(x) =

X

w(Ht(x))

Let x G Mi an d le t U b e a n ope n neighborhoo d o f x whic h i s the domai n o f a local char t 0 : U — • IRn, wher e n i s th e dimensio n o f M\. Le t ): f ow = fowocj)1 R. I f a G U is near x an d if a = (a) , we have: *((Ti) wher e G\ = dx\ A dx2 + dxs A dx± + h

dx2m-i A dx2m, (f

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t = 2m)

6

A. BANYAG A

in th e symplecti c case , an d G\ — dx\ A dx2 A • • • A dx n in th e volum e preservin g case . Th e existenc e o f thes e chart s ar e guarantee d b y Darboux' theorem . The vecto r fields dk = d/dxk define d o n U ca n b e extende d int o element s of C ai(Mi). Sinc e LQ ko\ — 0 o n U. Henc e i(dk)cFi i s a close d for m o n U. B y Poincare lemm a i t i s exact. Ther e exist s a functio n (symplecti c case ) o r a ( n — 2 ) form (volum e case ) (5k such tha t i(dk)(Ji = : U — » M 2 m + 1 i s such tha t ai\u i s a multipl e of *g_ where: a = dz - (y 1 dxi + y 2 dx 2 + • • • + y m dx m) Here (z, t/i, x i , . . ., y m , x m ) ar e coordinate s o n E 2 m + 1 . I t i s eas y t o chec k tha t th e following vecto r fields define d o n U ar e contac t vecto r fields: d

z = dz Ak = Bk =

d dxk d dyk

+ xk

d dz

To exten d thes e vecto r fields t o th e entir e manifol d int o contac t vecto r fields, w e take a smoot h functio n A lik e above , whic h i s 1 near x an d ha s suppor t i n U: thei r extensions Z , Ak, Bk ar e th e contac t vecto r fields r -1 (Ai(£)cri), wher e £ stand fo r one of the vector fields Z, Ak, Bk an d r i s the isomorphism sendin g a contact vecto r field X t o th e smoot h functio n i(X)a. Like in th e proo f o f Taken s theorem , w e can comput e th e partia l derivative s o f / o it?, the loca l expressio n o f / o w withi n th e canonica l chart s a s follows : i f x € U and [£ ] x Z x Z , where th e firs t facto r give s th e topologica l clas s associate d t o a give n Legendria n class an d th e secon d an d th e thir d factor s giv e the value s o f the invariant s r an d tb on thi s class . The inequalit y (Theore m 1 .2 ) an d th e congruenc e (Propositio n 1 .2 ) impos e restrictions o n the imag e o f the ma p A . However , thes e ar e ar e no t th e onl y restric tions. Additiona l restriction s wer e recently foun d b y Y. Kanda (se e [Ka]) , D. Fuchs and S . Tabachniko v (se e [FuTal] ) an d L . Rudolp h (se e [Ru]) . Our mai n theore m 1 . 1 state s tha t th e ma p A is injectiv e whe n restricte d t o A - 1 ([0] x Z x Z), where [0 ] denotes the topological class of the unknot. I t i s unknown (although seem s extremel y unlikely ) whethe r thi s remain s tru e fo r topologicall y non-trivial knot s i n the standar d contac t S 3. Fo r othe r manifold s (fo r instance , fo r M = (S 2 x 5 1 )#(S' 2 x S 1 )) th e ma p i s not injective , a s i t i s show n i n [F2] . Let D c Z x Z b e th e range of the invariants (r , tb) on the space of topologicall y trivial Legendria n knots . Eve n thoug h 1 . 2 an d 1 . 2 preclud e surjectivit y o f A o = A|[£]0, wher e [C]o = A _1 ([0] x Z x Z) , th e domai n V o f A o is a s larg e a s possibl e (see Fig . 6) . Indeed : LEMMA 1 .4 . V

= {(m , -\m\ -

2k - 1

)| k > 0 }

PROOF. Th e inequalit y 1 . 2 an d th e congruenc e 1 . 2 sho w tha t V c {(m , - | r a| - 2k - 1 ) | k > 0}. On th e othe r hand , fo r an y pai r (m , n) G V th e Fig . 6 provide s a wavefron t o f a Legendrian kno t L i n M 3 wit h tb(L) = n , r(L) = m . Sinc e an y contac t manifol d contains th e standar d contac t M 3 (b y Darboux' s theorem) , thi s show s tha t al l th e examples o f th e catalogu e ca n b e constructe d i n a genera l (M , £). • Catalog of Legendrian unknots. Figur e 6 provide s a lis t o f Legendria n wave fronts eac h lifting t o a Legendrian unkno t i n standard R 3 wit h al l admissible value s of (tb,r). Eac h (tb,r) cas e also include s a n indicatio n o f the characteristi c foliatio n that ca n b e obtaine d o n a spannin g dis k fo r th e lif t o f tha t front . 2. Manipulatio n o f characteristi c foliatio n The goa l o f thi s sectio n i s Lemm a 2. 5 below which , give n a topologicall y triv ial Legendria n knot , establishe s existenc e o f a spannin g dis c whos e characteristi c foliation ha s a specia l form , whic h w e call elliptic. W e will achiev e thi s b y alterin g its characteristi c foliatio n vi a gradua l deformatio n o f th e surface . Th e basi c tool s which will allow us to perform suc h manipulation s o f characteristic foliatio n ar e th e controlled birt h an d deat h o f elliptic-hyperboli c singularit y pairs . Th e controlle d death o f such a pair , i.e . it s killing, i s the subjec t o f Lemma 2.1 . I t wa s firs t prove d by E . Girou x i n [Gil] , an d i n a for m improve d b y D . B . Fuchs , wa s presente d i n [E7]. Th e creatio n o f a n elliptic-hyperboli c pai r o f singularitie s i s mor e straight forward an d i s eas y t o accomplis h i n general ; i n it s mos t basi c for m i t i s ofte n stated withou t proof . Here , however , i t wil l b e usefu l t o b e abl e t o perfor m suc h a pai r creatio n nea r a n existin g singularit y p i n suc h a wa y tha t certai n leave s o f the characteristi c foliatio n throug h p remai n fixed an d p itsel f appear s t o suffe r a Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

26

Y. E L I A S H B E R G AN D M . F R A S E R

tb= - ( 2 t + 1 + s ) r = -s

For r > 0 : reverse orientation on r < 0 diagram.

FIGURE

6 . Catalogu e o f wavefronts .

conversion betwee n ellipticit y an d hyperbolicity . Th e specific s o f this wil l b e give n shortly, i n th e Conversio n Lemm a 2. 2 whic h follow s th e Eliminatio n Lemm a 2.1 . LEMMA 2. 1 (Eliminatio n Lemma) . Let S be an embedded surface in (M,£ ) having just two singularities of the characteristic foliation, one that is elliptic p, and one that is hyperbolic q. Suppose that they are both of the same sign and that there is a separatrix 7 which connects q to p. Then given any arbitrarily small neighborhood of 7 , there exists a C°-small isotopy of S supported in that neighborhood and fixing 7 which results in a new surface having no singularities of its characteristic foliation (see Figure 7) . P R O O F . Se e [Gil ] o r [E7] . •

LEMMA 2. 2 (Elliptic-Hyperboli c Conversion) . Let S be an embedded surface in (M, £ ) having just one singularity of its characteristic foliation called p. Suppose p is elliptic (hyperbolic). Letr and a be two leaves of the characteristic foliation which pass through p, intersecting at this point only and transversally. Then given any arbitrarily small neighborhood of p, there exists a C°-small isotopy of S supported in that neighborhood and fixing r and a which results in a new surface having a Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

CLASSIFICATION O F T O P O L O G I C A L L Y T R I V I A L L E G E N D R I A N K N O T S 2

7

FIGURE 7 . Elliptic-hyperboli c elimination . hyperbolic (elliptic) singularity at the intersection of a andr and also two additional elliptic (hyperbolic) singularities, one on each side of p as shown in Figure S, all three singularities being of the same sign as p was and there being no other singularities on S. P R O O F . Se e [Fl] . •

Note tha t case s (b),(c ) i n Figur e 8 can b e summarize d b y th e observatio n tha t the newl y create d pai r o f ellipti c sin k (source ) singularitie s wil l o f necessit y b e created o n whicheve r o f a o r r flowe d int o (ou t of ) p. Fo r cas e (a ) w e hav e th e freedom t o choos e whic h o f the leave s a o r r wil l be grante d ne w singularities . Fo r all three cases , note tha t whil e we seem to creat e tw o elliptic o r els e two hyperboli c singularities, i n fac t w e ar e reall y jus t creatin g a n elliptic-hyperboli c pai r bu t on e of th e tw o newl y create d singularitie s take s th e forme r plac e o f p an d shove s p aside. Sinc e thi s i s don e whil e fixing r an d a however , i t i s perhap s bes t no t t o refer t o i t a s a shovin g asid e o f p. On e ca n certainl y not e tha t ultimatel y th e typ e of singularit y a t th e fixed poin t r f l cr is converte d (ellipti c eac h e + insid e attache d t o /i o n L. FIGURE +

we will consider wil l have all edges of the sam e type, eithe r al l singularity-free o r al l hyperbolic-containing. A mixed version will arise though i n the proof of Lemma 2.5. When reduce d for m ha s bee n achieved , on e ca n find a Legendria n tre e usin g positive ellipti c interio r singularitie s a s vertice s an d th e separatrice s throug h neg ative hyperboli c interio r singularitie s a s edges . Suc h graph s ar e discusse d i n [E7 ] where i t i s explaine d tha t the y mus t i n fac t b e trees . Note tha t tightnes s o f (M , £) implie s (derivatio n show n i n Fig . 1 0 ) tha t eac h positive ellipti c interio r poin t mus t b e connected t o at least one positiv e hyperboli c point o n th e boundary . Thi s fac t wil l b e importan t later . Legendrian tree s wil l b e centra l t o th e proo f o f th e Mai n Theore m i n th e final two section s o f th e paper , bu t th e tree s use d ther e wil l no t b e quit e lik e thos e i n [E7]; they will have singularity-fre e instea d o f hyperbolic-containing edge s (i.e . will be secon d o f tw o embeddin g type s define d earlier) . I n particular , ou r mai n focu s will b e Legendria n tree s tha t ar e embedde d i n a surfac e i n thi s manne r an d hav e the furthe r propert y tha t al l vertice s ar e elliptic . Suc h tree s automaticall y hav e alternately positiv e an d negativ e (elliptic ) vertice s an d involv e no hyperbolic point s at all . Thes e tree s wil l aris e ou t o f havin g wha t w e wil l cal l elliptic form instea d of reduced form fo r th e interio r characteristi c foliation . Recal l tha t afte r reduce d form ther e wa s t o b e a further , "cosmetic " manipulatio n o f foliation : w e wil l no w define an d the n conver t t o "ellipti c form" . Elliptic form on spanning disk. Le t D b e a spannin g dis k fo r th e Legendria n unknot L . Th e characteristi c foliatio n D^ wil l b e sai d t o b e i n elliptic form whe n the sign s o f boundar y singularitie s alternate , al l interio r singularitie s ar e ellipti c positive o r ellipti c negativ e and , beside s th e direc t connectio n t o it s neighbou r via a subsegmen t o f L , eac h boundar y poin t i s connecte d onl y t o interio r points , and moreove r eac h ellipti c interio r poin t i s connecte d t o a t leas t tw o boundar y hyperbolic points 5 . These condition s impl y tha t boundar y hyperboli c point s o f a give n sig n ar e connected t o eac h othe r i n group s o f tw o o r mor e vi a thei r separatrice s whic h meet i n a n ellipti c interio r poin t o f that sam e sig n an d thu s divid e th e surfac e int o regions o f th e for m (a ) o r (b ) o n whic h ther e ar e n o othe r interio r singularities , a s illustrated i n Figur e 1 1 . Thi s i s eas y t o se e onc e on e check s tha t th e basin (se e [E7]) o f a n ellipti c poin t i s alway s bounde d b y hyperboli c separatrice s an d s o i n 5 These boundar y hyperboli c point s ar e necessaril y o f sam e sig n becaus e /i + mus t b e sourc e of flow alon g L , henc e sin k o f flow fro m th e interio r an d vic e vers a fo r h~ .

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CLASSIFICATION O F TOPOLOGICALL Y TRIVIA L LEGENDRIA N KNOT S 3 1

(a) (b FIGURE

)

1 1 . Type s o f regio n o n dis k i n ellipti c form .

the cas e o f ellipti c form , wher e ther e ar e n o hyperboli c interio r points , th e basi n of a verte x v i s bounde d b y portion s o f L togethe r wit h th e separatrice s betwee n boundary hyperboli c an d interio r ellipti c points. Sinc e a boundary hyperboli c poin t can obviously b e attached t o only one interior point , thi s i s very restrictive. Indeed , when th e separatri x unde r consideratio n goe s t o v again , a type(b ) regio n results , when th e separatri x goe s t o a verte x othe r tha n v, a typ e (a) regio n results . Th e basin o f v i s the n th e unio n o f a t leas t tw o o f thes e regions ; Figur e 1 2 illustrate s this fo r severa l vertice s o f a spannin g disk . Th e detaile d argumen t i s lef t t o th e reader. When w e conside r jus t hal f o f a typ e (a) region , a s show n b y th e hatchin g i n Figure 1 1 , we will cal l i t a semi-type(a ) region . Skeletons and extended skeletons. Jus t a s reduced for m wit h norma l absorbin g boundary resulte d i n an obviou s Legendrian tre e o n the interior , s o too does ellipti c form. I t wil l be called a skeleton and, a s mentioned, wil l have singularity-free edges . To prepar e fo r th e definitio n o f th e skeleto n (an d extende d skeleton ) o f a dis k i n elliptic form , choos e on e Legendria n ar c betwee n eac h pai r o f connecte d ellipti c points. Not e tha t som e arc s g o betwee n interio r ellipti c points , other s betwee n a boundary ellipti c an d a n interio r ellipti c point . Now , le t the skeleton b e the Legen drian embeddin g o f a grap h tha t ha s fo r vertice s al l interio r ellipti c point s an d fo r edges th e abov e choices o f Legendria n segments . Le t th e extended skeleton b e th e Legendrian grap h embeddin g containin g th e skeleto n an d havin g al l th e boundar y singularities a s additiona l vertices , wit h correspondin g ne w edge s a s follows : be tween th e skeleto n an d th e boundar y ellipti c points , tak e th e abov e chose n arcs , between th e skeleto n an d th e boundar y hyperboli c points , tak e th e latter' s separa trices. Thi s i s illustrate d i n Figur e 1 3 . Not e tha t th e term s skeleto n an d extende d skeleton ma y b e use d variousl y t o designat e th e appropriat e Legendrian grap h em bedding o r th e correspondin g abstract graph . W e shal l se e i n th e nex t paragrap h that th e skeleto n a s a grap h i s i n fac t a tree . Hence , i t i s clearl y a Legendria n tree (o f type wit h n o mid-edge singularities ) becaus e o f the natur e o f characteristi c foliation o n th e dis k i n whic h i t arises . Similarl y fo r th e extende d skeleton . Skeleton uniqueness, simply-connectedness. Th e choic e o f a skeleto n i s combinatorially unique. Mor e precisely , th e se t o f edge-pair s o f vertice s i n th e skeleto n Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

32

Y. E L I A S H B E R G AN D M . F R A S E R

BASIN OF V F I G U R E 1 2 . Exampl e o f ho w basin s compose d o f typ e (a) an d type(b) regions .

Skeleton Extende

d Skeleto n

FIGURE 1 3 . Extendin g th e skeleton .

is uniquel y determine d (u p t o a globa l self-homeomorphis m o f th e characteristi c foliation, whe n suc h exists ) b y th e topolog y o f th e dis k foliation ; wher e a n edgepair refer s t o a pai r o f vertice s whic h ar e th e endpoint s o f som e edge . Indeed , since th e definitio n o f skeleton specifie s tha t th e skeleto n mus t includ e som e choic e of connectin g Legendria n ar c betwee n an y tw o vertice s wher e suc h a connectio n exists, i t specifie s i n othe r word s tha t whethe r o r no t a give n pai r o f vertice s i s t o be a n edge-pai r o f th e skeleto n depend s entirel y o n whethe r o r no t a Legendria n Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

CLASSIFICATION O F TOPOLOGICALL Y TRIVIA L LEGENDRIA N KNOT S 3

3

Cluster o f Positiv e Twig s FIGURE

1 4 . Exampl e o f dis k i n ellipti c form .

arc betwee n th e correspondin g tw o elliptic point s exists . Thi s i s clearly determine d entirely b y th e topolog y o f th e dis k foliation . Moreover, th e skeleto n mus t b e a connected graph. Thi s follow s readil y fro m the definitio n o f ellipti c for m a s follows . Assum e ther e i s mor e tha n on e verte x i n the skeleton (otherwis e graph triviall y connected ) the n tak e an y two interior ellipti c points A an d B. Eac h mus t hav e a t leas t tw o connection s t o boundar y hyperboli c points, an d L i s connected s o we can selec t on e suc h boundar y hyperboli c fo r eac h of A an d £? , sa y HA an d HB whic h ar e th e endpoint s o f som e subsegmen t o f L. We ca n the n follo w fro m HA t o HB th e successio n o f typ e (a ) region s tha t hav e an edg e o n ou r subsegment . Whe n tw o vertice s shar e a typ e (a ) region , the y ar e necessarily th e endpoint s o f an edge in the skeleto n an d s o the successio n o f region s gives a successio n o f edge s o f th e skeleto n whic h connect s AtoB. Note tha t thi s sam e correspondenc e betwee n subsegment s o f the boundar y an d paths i n the graph als o implies that th e graph is actually a tree: i f there were a cycle in th e graph , thi s woul d giv e u s a proper , non-trivia l componen t o f th e boundar y L whic h contradict s L bein g connected . Valence of vertex. W e will frequently b e dealin g wit h graph s s o let u s agre e o n a coupl e o f grap h theor y terms . Th e valence o f a verte x i n a n abstrac t grap h wil l be take n t o mea n th e numbe r o f edge s attache d t o tha t vertex . Whe n a verte x o f a tre e ha s valenc e on e i t wil l b e calle d a n end vertex. Positive/negative twigs and clustering. Th e Legendria n ar c betwee n a positiv e (negative) verte x and a boundary hyperboli c point o f same sign will be referred t o a s a positive (negative) twig. Conside r th e twig s fo r exampl e a t poin t p i n Figur e 1 4) . When mor e tha n tw o twig s ar e attache d t o a give n vertex , i t wil l b e calle d a cluster vertex. Suc h clustering at positive p is a feature o f examples with r ^ — 1. O n the othe r hand , clusterin g a t negativ e p occur s whe n r > — 1. I n bot h thes e cases , clustering o f th e opposit e sig n ma y als o occur , bu t i s no t guaranteed . Similarly , when r = — 1, no clustering nee d occu r though i t might . I n a sense, r thu s measure s Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

34

Y. ELIASHBER G AN D M . FRASE R

Edges ar e numbere d t o indicat e a cyclical orderin g . . . cr , r fo r exampl e are adjacent . FIGURE

1 5 . Whe n ar e tw o edge s adjacent ?

^^ 2 .

how man y boundar y singularitie s ar e i n exces s o r ar e wanting . Whe n clusterin g o f both sign s occurs , w e wil l ai m t o cance l al l nonessentia l clustering , i.e . reduc e t o clustering o f just on e sign. Ho w it i s that th e sig n of an essentia l cluste r indicate s a shortage o r exces s o f boundar y singularitie s an d ho w r influence s thi s wil l bot h b e explained i n th e paragrap h "Ho w r influence s clustering " a t th e en d o f the presen t section. Reduced form — > elliptic form. I n th e nex t lemma , w e wil l b e especiall y con cerned wit h vertice s o f valenc e greate r tha n 2 an d th e edge s tha t ar e attache d t o such vertices . W e coul d cyclicall y orde r th e attache d edge s a t suc h a verte x sim ply b y usin g th e surfac e orientatio n o r it s opposit e a s ou r convention . I n eithe r case, suppos e tw o edge s attache d t o th e verte x p ar e adjacen t wit h respec t t o th e ordering, the n le t u s refe r t o the m a s adjacent edges 6. Figur e 1 5 illustrates this . Vertices wil l als o b e referre d t o a s adjacent sometime s bu t thi s wil l b e i n th e familiar sens e o f bein g th e tw o endpoint s o f som e edge . LEMMA 2.5 . In tight (M,£) , let D be an embedded disk spanning the unknot L. Suppose D^ is in reduced form with normal absorbing boundary. Then there is a C°-small isotopy of D fixing L which results in a new spanning disk having characteristic foliation in elliptic form. P R O O F . W e ar e assumin g D^ i s reduce d s o i t ha s a Legendria n tre e T. Le t p be a vertex of T havin g valence greater tha n 2 , if such exists. Suppos e tha t betwee n every tw o adjacen t edge s a t p , there i s a connectio n fro m p t o a positiv e boundar y hyperbolic poin t a s shown i n Figure 1 6 (a) , then d o nothing. Suppose , o n the othe r hand tha t thi s i s no t th e case ; se e Figur e 1 6 (b) . In thi s case , perfor m th e manipulatio n specifie d i n Figur e 1 7 which i s a com position o f elliptic-hyperboli c eliminatio n (GE ) an d elliptic-hyperboli c conversio n (EHC), henc e ca n b e mad e arbitraril y C°-small . Thi s manipulatio n will replac e p 6

T h e n this , o f course , doe s no t depen d o n whic h convention—positiv e o r negative—w e chose .

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CLASSIFICATION O F T O P O L O G I C A L L Y T R I V I A L L E G E N D R I A N K N O T S 3

5

convert each e- t o h- usin g EHC

kill 2 (e-, h-) pair s (—) usin g GE

FIGURE

1 7 . Alterin g foliatio n a t a for k i n tree .

and an y adjacen t interio r hyperboli c point s b y a singl e interio r e~ , whic h w e cal l q. Le t u s pic k a singl e Legendria n ar c connectin g q to eac h positiv e ellipti c poin t that wa s formerl y adjacen t t o p i n T . The n conside r th e grap h wit h thes e arc s a s edges an d thes e point s plu s q as vertices. I t i s a Legendria n tree 7 wherei n q has th e same valenc e a s p had . Bu t i t ha s th e advantag e tha t i t i s mor e symmetri c a t thi s point; specifically , betwee n ever y two adjacen t edge s a t q there i s a connectio n t o a boundary hyperboli c poin t o f sam e sign , i.e . i n thi s cas e a n h~. Once the abov e procedure ha s bee n performe d a t al l eligible vertices, we will b e able t o conclud e tha t al l interio r ellipti c point s have , betwee n an y pai r o f adjacen t edges, a n attachmen t t o a same-sig n boundar y hyperboli c point . A t thi s stage , w e

7

w i t h singularity-fre e edges .

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36

Y. E L I A S H B E R G AN D M . F R A S E R

have a Legendrian tre e wit h edge s that ar e variously singularity-fre e o r hyperbolic containing. Tha t mean s tha t som e vertice s ma y b e negative , other s positive . Le t us cal l thi s mixe d Legendria n tre e S. Recall tha t ever y positiv e ellipti c interio r poin t i s connecte d t o a t leas t on e positive hyperboli c boundar y poin t b y tightness . Next , w e would lik e ever y verte x to hav e a t leas t tw o connection s t o same-sig n boundar y hyperboli c points . Cer tainly, an y negativ e vertice s tha t hav e bee n create d b y th e previou s ste p hav e thi s property. Bu t som e o f th e ol d positiv e vertice s ma y not . Suppos e p i s a (necessar ily positive , an d valenc e < 2 ) verte x wit h onl y on e boundar y (positive ) hyperboli c connection. Le t M b e th e maxima l connecte d subtre e o f S containin g p an d al l of whose vertice s hav e thi s deficiency . Sinc e al l M' s vertice s hav e valenc e < 2 , M i s linear. O n eac h en d o f M (o r eac h 'side ' i f M — p) ther e i s a negativ e singularit y that i s connected t o a t leas t tw o negative boundar y vertices . Conside r th e regio n 1Z formed b y "cutting " acros s D throug h eac h o f thes e negativ e interio r singularities . By performin g elliptic-hyperboli c eliminatio n an d elliptic-hyperboli c conversio n a s indicated i n Figure 1 8 , we eliminate al l but on e interio r singularit y fro m th e regio n 1Z] i t wil l b e a negativ e ellipti c poin t attache d t o (on e o r more ) e + an d (tw o o r more) h~ boundar y point s (a s show n i n Figur e 1 8) . Let u s assum e tha t th e (C°-small ) manipulatio n o f Figur e 1 8 ha s bee n per formed o n al l 'deficient ' positiv e vertices . Now , al l vertice s wil l hav e a t leas t 2 connections t o same-sig n boundar y hyperboli c points . Whereve r ther e i s a n inte rior (negative ) hyperboli c point , le t u s appl y th e alteratio n indicate d i n Figur e 1 9 . Like bot h previou s alterations , i t to o i s jus t a compositio n o f hyperbolic-ellipti c conversion an d elliptic-hyperboli c elimination , an d a s suc h ca n b e chose n arbitrar ily C°-small. It s effect i s to convert th e interio r hyperboli c poin t t o an elliptic point , and th e tw o boundar y point s t o whic h i t wa s attache d t o hyperboli c ones . At thi s stage , th e dis k foliatio n i s i n ellipti c form . Indeed : sig n alternatio n on th e boundar y ha s bee n retaine d throug h eac h o f th e 3 manipulations ; s o to o has th e propert y tha t boundar y singularitie s connec t onl y t o thei r direc t boundar y neighbors o r interior points ; n o hyperbolic interio r singularitie s remain ; an d finally, the interio r (elliptic ) point s o f bot h sig n ar e connecte d t o a t leas t tw o boundar y hyperbolic point s each . • How r influences clustering. Usin g th e formul a

e±-hT =

±(lTtb±r)

quoted earlie r (Lemm a 2.4 ) i t i s easily checke d tha t ther e wil l be mor e tha n 2(e + + h- + 1 ) boundar y vertice s whe n r < — 1 an d s o b y th e pigeonhol e principl e r < — 1 implies that , i n reduce d foliation , som e interio r positiv e ellipti c poin t mus t b e connected t o more than 2 positiv e hyperboli c point s o n th e boundar y an d hence , when th e manipulatio n jus t describe d i s applied , thi s poin t wil l ge t converte d t o a positive cluste r vertex . Th e sam e reasonin g fo r r > — 1 means, i n reduced foliation , some interio r positiv e poin t wil l b e connecte d t o less than 2 positiv e hyperboli c points o n th e boundary , an d thu s afte r th e foliatio n i s altere d a s indicate d i n th e previous proof , will b e connecte d t o a negativ e cluste r vertex . A t eac h vertex , le t us refe r t o an y twig s beyon d th e first tw o a s extra twigs. I n Sectio n 3.2 , w e will se e how extr a twig s o f opposit e sig n ca n b e "cancelled" , thu s refinin g th e connectio n between r an d twigs : w e will b e abl e t o coun t th e numbe r o f essential extr a twig s on th e basi s o f th e invarian t value s (tb,r).

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CLASSIFICATION O F T O P O L O G I C A L L Y T R I V I A L L E G E N D R I A N K N O T S

n intermediate h+ h+ e- h + e - h + h

+ We start here with n deficient positive e+ vertice s in a row

jEHC h+h-h+ h - h + h +

h+ h- h+ h- h+ h +

h+ .

EHC T

h + n intermediate h+

*

h+ h-e + h

eh-

'

- e + h+

e+ h+

Now we're done: we've created a negative vertex with n extra negative twigs (i.e. a total of n+2 negative twigs)

F I G U R E 1 8 . Convertin g a ro w o f 'deficient' positiv e vertice s t o a negative vertex .

(Elliptic-Hyperbolic Conversion ) (Eliminatio

FIGURE

n Lemma )

1 9 . Convertin g Negativ e Points : hyperboli c 2 a t interio r positiv e (negative ) ellipti c points . Le t T b e a choic e o f skeleto n o n th e dis k an d le t u s tak e som e embeddin g T o f T i n R 2 suc h that th e slop e o f ever y edg e i s betwee n ±e . Le t u s als o attac h smal l segment s a t each verte x o f T i n E 2 , positione d i n suc h a wa y a s t o replicat e th e arrangemen t of twigs o n th e skeleto n T o f D, bu t withou t enforcin g an y restriction s o n th e seg ments' slope s except th e following : sinc e each vertex of T ha s at leas t tw o twigs, le t 8

This Legendria n isotop y i s obtaine d b y composin g th e isotopie s betwee n eac h Legendria n knot an d th e th e commo n catalo g knot .

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CLASSIFICATION O F T O P O L O G I C A L L Y T R I V I A L L E G E N D R I A N K N O T S 3

9

us als o ensure tha t a t eac h vertex o f T, w e position tw o twigs vertically . Moreover , assume eac h verte x o f T ha s a sig n correspondin g t o th e sig n o f th e ellipti c poin t it represents . Let u s recor d thi s informatio n a s show n i n Figur e 20 . Suc h a diagra m will b e referred t o a s a twigged tree. Not e that a twigged tre e i s thus an y abstrac t tre e plu s twigs, wit h a sig n o n on e o f it s vertice s (sign s wil l the n alternate) , an d such that every vertex has at least as many twigs as valence and the twigs appear between all adjacent edges. Thi s las t conditio n i s crucial to bein g abl e to construc t a wavefron t from a twigge d tree . I t i s a propert y automaticall y possesse d b y a twigge d tre e that ha s com e fro m a dis k i n ellipti c form . Thi s i s due t o th e fac t tha t suc h a dis k can b e decompose d int o (a ) an d (b ) typ e region s a s mentione d i n th e paragrap h o f Section 2 entitled "Ellipti c for m o n spannin g disk" . $

felcpe)< 6 -y,

Ell. for m Twigge FIGURE

d Tre e Proj

. Twigge d Tre e

20 . Variou s record s o f dis k foliation .

FRONT CONSTRUCTION ALGORITHM

FIGURE

21 . Twigge d tre e— > wavefront.

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40

Y. E L I A S H B E R G AN D M . F R A S E R

Given a twigge d tree , i f i t i s embedded i n R 2 a s describe d abov e fo r skeletons , then le t u s cal l i t a projected twigged tree (se e Figur e 20) . Now, based o n the record o f D^s structur e tha t i s stored i n a projected twigge d tree, a wavefront wil l be constructed. I n general: i f T i s a projected twigge d tree, we shall denot e b y WT th e fron t constructe d i n th e algorith m below ; i f th e projecte d twigged tre e use d i n th e algorith m belo w ha s bee n obtaine d fro m a n elliptic-for m spanning surfac e fo r th e Legendria n kno t L , w e shal l us e th e notatio n W(L) fo r the fron t an d wil l be carefu l t o mak e th e particula r spannin g surfac e involve d clea r from th e context . Front construction algorithm. Le t u s assum e no w tha t w e hav e a projecte d twigged tree . I n cas e i t aros e fro m th e skeleto n o f som e disk , w e wil l nee d t o be vigilan t o f th e cycli c orderin g o f th e variou s semi-typ e (a ) region s whic h i s induced b y th e orientatio n o f th e disk' s boundary . I n orde r t o accomplis h this , it will b e usefu l t o chang e ou r projecte d twigge d tre e int o a ne w on e (fo r th e rest o f thi s constructio n only) : revers e th e relativ e position s o f edge s an d twig s t o the righ t o f eac h positiv e verte x an d t o th e lef t o f eac h negativ e vertex . Thi s i s demonstrated, alon g wit h th e res t o f th e construction , i n Figur e 21 . No w procee d with th e algorithm : 1. A t eac h positive or negative en d vertex, plac e respectively th e first o r secon d piece of wavefront (whic h will "wrap " aroun d th e "ope n side " o f the vertex) :

POS:

< — € ^o

2. No w a t ever y non-en d verte x wit h exactl y tw o twigs , plac e a crossin g o f the wavefron t wit h upwar d orientatio n i n th e positiv e cas e an d downwar d orientation i n th e negativ e case :

X^V^NEG

POS: tfrtH

: ^

3. Finally , fo r ever y additiona l twi g (beyon d th e first two ) a t eac h vertex , ad d a zig-za g to th e fron t a t tha t positio n (usin g PO S o r NE G typ e show n whe n vertex i s respectively positiv e o r negative) . Thi s shoul d resul t i n the over-al l effect show n t o th e right .

f^fc Once al l vertices i n th e tre e hav e ha d on e of these thre e buildin g block s drawn o n to p o f th e projecte d twigge d tre e a t th e correspondin g place , w e simply attac h eac h on e t o th e adjacen t ones : Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

CLASSIFICATION O F TOPOLOGICALL Y TRIVIA L LEGENDRIA N KNOT S 4 1

We obtain a wavefront fo r the entire projected twigge d tree, i.e. for the structur e of D^. A n exampl e i s given i n Figur e 21 . Clearly th e constructio n algorith m ma y b e performe d i n reverse, allowin g u s t o reconstruct th e appropriat e twigge d tre e fro m a fron t tha t i s composed o f buildin g blocks o f th e type s use d i n th e algorithm . Front s tha t ar e so-compose d will b e called universal fronts. Therefor e w e hav e show n a 1 - 1 correspondenc e betwee n twigged tree s an d universa l fronts . Recal l tha t th e value s o f (tb, r) ca n b e rea d of f from eithe r a twigge d tre e o r universa l front . Thi s i s show n i n Figur e 22 . It i s als o importan t t o not e tha t give n a certai n twigge d tree , th e Legendria n lift o f the fron t thu s constructe d ca n b e give n a spannin g dis k whos e foliatio n i s i n elliptic for m an d i n whic h ther e i s th e sam e twigge d tre e a s originall y used . Th e whole proces s i s thus closed . 3.2. Simplifyin g fronts . Thi s portio n o f th e pape r wil l attemp t t o simplif y the front s w e hav e created . Specifically , w e wil l sho w tha t an y fron t obtaine d b y the abov e algorith m ca n b e transversall y homotope d t o on e of the front s presente d in the earlie r catalog of wavefronts. Not e tha t a transversal homotop y o f fronts lift s to a Legendria n isotop y betwee n th e correspondin g Legendria n knot s t o whic h th e fronts lifted . Indeed , thi s follow s fro m th e fac t tha t w e ca n canonicall y lif t eac h transitional fron t i n th e homotop y o f front s an d thi s lif t ( a Legendria n knot ) wil l be embedded i f each self-intersectio n poin t o f the correspondin g fron t i s transversa l (recall fro m Sectio n 1 .2) . Let a "zig-zag " denot e th e thir d typ e o f componen t mentione d i n th e fron t construction algorithm . Then :

•h^w. •&= -[-t

Universal Fron t FIGURE

+ t-v]

=

Twigged Tre e 22 . Determinin g (tb,r) fro m fron t o r tree .

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-if

42

Y. E L I A S H B E R G AN D M . F R A S E R

LEMMA 3.1 . There is a transversal homotopy which moves a zig-zag t o any position fro m any position on the universal front. PROOF. I n th e following , 'move ' wil l mea n 'displac e b y transversa l isotopy' . ZIG-ZAG ALGORITHM .

1. On e ca n mov e a zig-za g aroun d an y en d o f th e

front.

^


^>>- shapes minu s the total numbe r o f -< and >shapes i n the front. Figur e 2 4 shows thi s calculatio n an d also tha t i t is equivalent, when T — WT to calculating th e size (i.e . #vertices) o f the tree minu s th e total number o f twigs. The maxima l valu e for the size of a (tb,r) wavefron t i s easily see n to be r +1 + \r + 1 | — tb. Not e tha t front s fro m th e catalog ar e special i n that the y ar e linear; each on e has twigs o f only on e sign, an d these twig s li e all at one end of the front . Also not e tha t th e catalog's fron t fo r a Legendrian kno t wit h invariant s (tb,r) ha s size r + l - f | r - f - l | — tb, the maximal size . An algorithm for transversally homotopin g a universal front t o the catalog fron t of the sam e (tb, r) value follows. Thi s algorith m use s the strategy: i f the underlyin g Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

44

Y. E L I A S H B E R G AN D M . F R A S E R

twigged tree is not linea r and/o r ha s twigs of both signs , then i t can be transversall y homotoped t o anothe r fron t wit h strictl y greate r size . Th e algorith m prove s th e following lemma : LEMMA 3.2 . Any front can be transversally homotoped to the front in the catalog of wavefronts that has the same value of (tb,r). If the original front was not already linear with all extra twigs of one sign, then this process will strictly increase the size of the front.

As a corollar y o f thi s lemma , w e thus obtain : COROLLARY 3.1 . Any front whose size is maximal (given its value of (tb,r)) must be linear and have extra twigs of only one sign. It will differ from a catalog front at most by the placement of the extra twigs.

Throughout th e following , 'front ' wil l mea n an y give n universa l front , an d 'move' wil l onc e agai n mea n 'displac e b y transversa l isotopy' . Ther e wil l b e tw o claims ( A an d B ) the n a mai n algorithm . Th e claim s ar e t o establis h step s use d i n the mai n algorithm . CLAIM A . / / the front has extra twigs of both signs, then there is a transversal homotopy of the front which gives a new front whose corresponding twigged tree is exactly the same except that two preselected twigs—one of each sign —have been eliminated. This move increases the size of the front by 3.

PROOF O F CLAIM A . Le t th e tw o chose n twig s b e move d b y Lemm a 3. 1 s o they bot h li e a t a n en d o f th e front . Suppos e th e en d verte x i s calle d v. The n th e move:

results i n th e eliminatio n o f th e zig-zags , henc e decreas e o f 2 in th e tota l # twigs , and i n th e creatio n o f a ne w vertex , henc e increas e o f 1 in th e lengt h o f th e tree . Thus th e siz e i s increase d b y 2 + 1 = 3 . • CLAIM B . If the underlying tree of a front whose extra twigs lie all at one end is not linear, then there is a front move which increases the size of the front by 1 . P R O O F O F CLAI M B . I n [E7] , linearizatio n o f a (different ) typ e o f tre e wa s accomplished b y constructin g an d the n transversall y homotopin g wavefronts . But , a fron t withou t twig s i s easily checke d t o b e o f exactl y th e sam e kin d a s th e front s used i n [E7] . And , sinc e th e fron t move s i n [E7 ] ar e performe d usin g onl y on e en d of th e tree : the y ma y therefor e b e use d her e o n front s whos e extr a twig s li e al l a t one end . W e repea t thi s algorith m below i n th e for m adjuste d t o ou r situation . Let W b e a universa l fron t withou t twigs . Her e i s th e algorith m consistin g o f front move s whic h brin g th e underlyin g tree s close r an d close r t o linearity . 1) I f tre e i s no t linea r the n som e verte x ha s valenc e greate r tha n 2 , i.e . ther e exists a "fork " i n th e tree . B y constructio n o f front s fro m trees , ther e mus t therefore b e a portio n o f th e fron t whic h look s a s follow s an d henc e w e

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CLASSIFICATION O F TOPOLOGICALL Y TRIVIA L LEGENDRIA N KNOT S 4

5

can perfor m th e indicate d fron t mov e (not e fron t gain s tw o crossings , i.e . conventional lengt h o f tree increases b y 1 , so for us 9 : lengt h of tree increase s by 2) . 2) I f th e tre e i s not linear , repea t par t 1 ) . Note that th e maximal length of a tree representing the Legendrian unknot wit h invariants (tb,r) i s — tb + r, wher e we count bot h positiv e an d negativ e singularitie s as vertices . Th e conventiona l tre e lengt h (countin g positiv e vertice s only ) woul d have bee n \{\ — tb + r) . • P R O O F O F LEMM A 3.2 . Th e comment s precedin g Clai m A and B apply to thi s algorithm a s well : assum e w e have an y give n universa l front . No w procee d with : Front simplification algorithm. 1. Repea t th e mov e o f the Clai m A unti l extr a twig s o f onl y on e sig n remain . 2. Whe n ther e ar e extr a twig s o f only on e sign, mov e al l of these t o on e end b y Lemma 3.1 . 3. Whe n th e fron t ha s twig s a t onl y on e end , us e Clai m B t o lineariz e th e underlying tree . By th e siz e remark s a t th e en d o f eac h claim , w e see tha t th e procedur e o f th e algorithm will strictl y increas e th e siz e o f th e front , unles s th e fron t wa s alread y linear wit h extr a twig s al l o f on e sign . •

The followin g lemm a summarize s wha t w e achieve d s o far . LEMMA 3.3 . For any topologically trivial Legendrian knot L in a tight contact manifold (M , £) there exists a spanning disk D whose foliation D^ is in elliptic normal form. Moreover, for the unique catalog knot with the same value (tb,r) there exists a spanning disc with characteristic foliation diffeomorphic to D^.

PROOF O F LEMMA 3.3 . Th e transversal homotop y o f fronts i n Lemma 3. 2 lift s to a Legendria n isotop y betwee n th e correspondin g Legendria n knots . Give n an y Legendrian unkno t L , w e sa w i t ca n b e spanne d b y a dis k i n elliptic-for m an d that ther e i s a universa l fron t whos e lif t L ha s a spannin g dis k wit h thi s 'same ' (i.e. diffeomorphic ) foliation . B y Lemm a 3.2 , then , L i s Legendria n isotopi c t o th e catalog kno t tha t ha s th e sam e (tb,r). Thi s isotop y lift s t o a n ambien t contac t isotopy b y Lemm a 1 .1 , an d s o w e ca n spa n th e catalo g kno t wit h a dis k tha t ha s characteristic foliatio n diffeomorphi c t o tha t o f L an d L (b y takin g a s ou r dis k th e image o f the spannin g dis k fo r L unde r th e fina l contactomorphis m i n th e ambien t contact isotopy) . • 4. En d o f th e proo f o f Mai n Theore m In thi s sectio n w e conclud e th e proo f o f Mai n Theore m 1 .1 . Having establishe d Lemm a 3. 3 th e proo f coul d b e finishe d usin g th e one parametric versio n o f th e classificatio n theore m fo r tigh t contac t structure s fro m [E6] (compar e ou r argumen t i n Sectio n 5 for overtwiste d contac t structures) . Thi s 9 Recall ou r tree s hav e singularity-fre e edge s bu t vertice s o f bot h signs . Thu s whe n th e front i s lengthene d b y tw o crossings , w e wil l conside r a spannin g surfac e whos e skeleto n ha s bee n lengthened b y th e additio n o f a negativ e ellipti c an d a positiv e ellipti c point , instea d o f negativ e hyperbolic an d positiv e ellipti c a s woul d hav e bee n don e fo r th e conventiona l tree s i n [E7] .

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46

Y. ELIASHBER G AN D M . FRASE R

was on e o f th e way s o f reasonin g explore d i n [Fl] 1 0 . However , th e require d 1 parametric versio n o f th e result s fro m [E6]—thoug h formulate d i n [E6]—i s no t proved there . Thu s w e decide d t o conclud e her e th e proo f o f th e Mai n Theore m directly, withou t referin g t o th e result s fro m [E6] . In vie w o f Lemm a 3. 3 th e proo f o f Theore m 1 . 1 wil l b e conclude d i f w e prov e the followin g LEMMA 4.1 . Suppose Legendrian knots L and V bound D and D' with diffeomorphic characteristic foliations in elliptic normal form. Then L and L' are Legendrian isotopic.

A simila r statemen t wa s establishe d i n [E7 ] fo r transversa l knots . Recal l tha t in th e proo f o f th e transversa l classificatio n theore m i n [E7] , there wa s a n obviou s transversal isotop y tha t wa s writte n righ t i n th e surface' s characteristi c foliation , and tha t allowe d u s t o shrin k th e transversa l boundar y t o arbitraril y clos e t o it s interior Legendria n tree ; then , becaus e thes e tree s wer e homotopi c vi a a n ambien t contact isotop y (comp . Lemm a 1 . 2 i n Sectio n 1 . 4 o f thi s paper) , thi s provide d a contact isotop y between the two transversal knots . Ther e is no such obvious isotop y in th e Legendria n case , bu t i n a n attemp t t o emulat e thi s metho d o f proof , le t u s make th e followin g definitio n an d constructio n (a s wa s don e i n [Fl]) . Adjusting spanning discs near their extended skeletons. Becaus e th e extende d skeletons T an d T " of the tw o ellipti c disc s S an d S' ar e diffeomorphi c ther e exists , according t o Lemm a 1 . 2 a globa l contac t isotop y takin g T' t o T. Th e sam e lemm a also allow s u s t o assum e tha t a smal l piec e o f V aroun d eac h ellipti c en d verte x of T" (i.e . en d verte x o f th e skeleton ) i s mappe d t o a correspondin g piec e o f L. Thi s isotopy map s L' t o a Legendria n isotopi c kno t an d S' t o a n ellipti c spannin g dis c for tha t knot . Thu s w e ma y a s wel l assum e th e tw o knot s L an d V hav e ellipti c spanning disc s S an d S' whic h shar e a commo n extende d skeleton , an d als o that L and L' themselve s coincid e nea r th e endpoint s o f the skeleton . The surface s S an d S' ar e tangen t a t th e finite se t A o f al l vertice s o f th e common extende d skeleto n T = X" , and thei r orientation s a t thes e point s coincide . Thus i t i s possibl e t o C 1 -perturb S' i n a smal l neighborhoo d o f th e se t A i n orde r to mak e th e ne w S' coincid e wit h 5 i n a (usuall y smaller ) neighborhoo d U D A. This alteratio n ca n b e don e keepin g th e characteristi c foliatio n o n S' i n a n ellipti c form, an d moreove r homeomorphi c t o th e characteristi c foliatio n o n 5 . A secon d type o f alteratio n ca n no w b e performe d whic h make s S' coincid e wit h S al l alon g the commo n extende d skeleton , an d no t jus t aroun d th e vertices . Indeed , notic e that fo r eac h Legendria n ar c r o f th e skeleto n T th e surfac e S f ca n b e rotate d i n a neighborhoo d o f r \ U int o th e surfac e S i n suc h a wa y tha t durin g th e whol e deformation th e topolog y o f the characteristi c foliatio n o n S' remain s unchanged . As a result o f these tw o alterations w e can assum e tha t th e Legendria n knot s L and V spa n elliptic-for m disc s S an d S' whic h coincid e i n a neighborhoo d o f thei r entire commo n extende d skeleto n T . Notic e tha t th e characteristi c foliation s o n S and S' remai n homeomorphic , an d wit h som e additiona l care , ca n b e eve n mad e diffeomorphic, bu t thi s i s irrelevant fo r ou r purposes . Exceptional spanning surfaces. Give n a topologically trivia l Legendria n kno t L with r < 0 , a n exceptional spanning surface (abbrev . ESS ) fo r i t i s a surfac e wit h 10

Essentially, [Fl ] introduce s th e ide a o f rephrasin g th e classificatio n proble m fo r unknot s i n terms o f th e classificatio n o f tigh t contac t structure s o n th e complemen t o f a ball ; th e latte r ca n then b e solve d usin g a versio n o f result s fro m [E6] , a s mentioned .

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CLASSIFICATION O F TOPOLOGICALL Y TRIVIA L LEGENDRIA N KNOT S 4

7

{Z = 0 }

FIGURE

25 . Mode l fo r singularit y curve .

characteristic foliatio n o f th e typ e show n i n Figur e 25 . I t involve s th e non-generi c (i.e. exceptional ) phenomeno n o f singularit y curves ; i.e . Legendria n curve s (withi n the characteristi c foliatio n o f a surface ) consistin g entirel y o f singularitie s o f th e characteristic foliatio n o f that surface . A small neighborhoo d o f such a line/surfac e pair ca n b e modele d b y a neighborhoo d o f th e lin e y = z = 0 on th e surfac e z = 0 in standar d M 3. Note tha t al l th e singularitie s i n a singularit y curv e ar e o f th e sam e sign , an d we can thu s spea k o f positive o r negativ e singularit y curves . B y slightl y perturbin g the surfac e z = 0 w e alte r th e characteristi c foliatio n i n suc h a wa y a s t o conver t the positiv e singularit y curv e t o a Legendria n curv e bearin g alternatin g hyperboli c and ellipti c positiv e singularities . Th e sam e woul d o f cours e wor k fo r a negativ e example. Also , w e ma y accomplis h th e reverse , thereb y creatin g a ±v e singularit y curve fro m a sequenc e e ±, h ±, e ± , h ±. Th e mean s o f doin g thi s i s a versio n o f Elliptic-Hyperbolic Eliminatio n wher e instea d o f twistin g fa r enoug h t o kil l th e singularities alon g a curve , w e twist jus t til l wher e th e surfac e become s tangen t t o £ everywhere alon g tha t curve . A spannin g surfac e S fo r a Legendria n kno t K wil l b e calle d exceptional whe n its characteristi c foliatio n i s i n norma l absorbin g for m alon g K, bu t it s interio r foliation ha s bee n obtaine d b y startin g wit h ellipti c for m an d convertin g al l Leg endrian curve s connectin g sam e sig n singularitie s int o singularit y curves , followin g the metho d indicate d above . Because thi s constructio n start s fro m ellipti c form , i t give s ris e t o sector s o f exactly th e typ e show n i n Figur e 26 . Note tha t i n genera l suc h a surfac e canno t b e C 2 -smooth becaus e o f th e clus tering a t p see n i n case s lik e tha t o f Figur e 1 4 . Suc h clusterin g i s a featur e o f examples wit h r ^ 0 , a s explaine d i n th e paragrap h 'Ho w r influence s clustering ' of Sectio n 2 . P R O O F O F LEMMA 4.1 . Above , we constructed elliptic-for m spannin g surface s S an d S' fo r L an d V whic h coincid e i n a neighborhoo d o f their extende d skeleton . Let u s conver t eac h o f thes e surface s int o a n ESS , a s describe d i n th e previou s Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

48

Y. E L I A S H B E R G A N D M . F R A S E R

+(-)

FIGURE

26 . Secto r betwee n tw o singularit y lines .

FIGURE 27 . Th e isotop y writte n o n a n ESS .

paragraph, takin g car e t o perfor m th e conversio n identicall y wher e S an d 5 ' coin cide s o a s t o obtai n ESS' s whic h als o coincid e i n a neighborhoo d o f th e commo n skeleton. Le t u s cal l th e resultin g ESS' s agai n S an d S" . Writte n righ t o n eac h surface, ther e i s a n isotop y supporte d i n th e complemen t o f smal l neighborhood s of th e en d vertice s whic h bring s th e portio n o f th e Legendria n kno t outsid e thes e neighborhoods arbitraril y clos e to th e skeleton . Thi s i s indicated i n Figur e 27 . Suppose tha t th e relevan t portio n o f L an d als o o f V ar e eac h isotope d fa r enough tha t the y reac h th e regio n aroun d th e commo n skeleto n wher e S an d S f coincide. B y applyin g th e Ellipti c Pivo t Lemm a (i.e . Lemm a 1 .3) , w e ma y exten d each o f thes e isotopie s t o th e neighborhood s o f en d vertices . W e thu s obtai n Leg endrian isotopie s takin g L an d V t o th e sam e Legendria n kno t (whic h i s T outsid e of th e en d verte x neighborhoods) . Thu s L an d V ar e Legendria n isotopic . Thi s concludes th e proo f o f Lemma 4.1 , an d wit h it , th e proo f o f Main Theore m 1 .1 . •

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CLASSIFICATION O F TOPOLOGICALL Y TRIVIA L LEGENDRIA N KNOT S 4

9

5. Legendria n knot s i n overtwiste d contac t 3-manifold s We sho w i n thi s sectio n tha t th e proble m o f classificatio n o f Legendria n knot s and link s i n overtwiste d contac t manifold s i s essentiall y simple r i n vie w o f classifi cation o f overtwiste d contac t structure s give n i n [E4] . Let (M , f) b e a n overtwiste d contac t manifold . A Legendria n kno t (o r link ) L c M i s calle d loose i f th e restrictio n o f contac t structur e f t o th e complemen t M\L i s still overtwisted. Otherwise , i f f| M\ L i s tight w e will call L exceptional. I n fact w e d o no t kno w an y example s o f exceptiona l Legendria n knot s i n overtwiste d contact manifolds . S o it coul d b e tha t al l Legendria n knot s i n overtwiste d contac t manifolds ar e loose . EXAMPLES. 1

. Suppos e that (M , f) i s a non-compact manifol d overtwisted at infinity. 1 1 The n an y Legendria n lin k L c M i s loose . 2. Suppos e tha t a homologica l t o zer o Legendria n kno t L i n (M , f) violate s Bennequin's inequalit y (se e Theore m 1 . 2 above) . The n L i s loose . For simplicity , w e conside r her e onl y th e cas e M = S 3. I t i s no t difficul t t o prove similarl y th e classificatio n resul t fo r loos e Legendria n knots , o r eve n links , i n any overtwiste d contac t manifold . THEOREM 5.1 . Let LQ, LI be two loose Legendrian knots in an overtwisted contact (5 3 ,f). Suppose that LQ and L\ are isotopic as smooth knots, and have the same values of the invariants tb and r. Then LQ and L\ are Legendrian isotopic. REMARK. A simila r resul t wa s independentl y obtaine d b y Fuch s an d Tabach nikov (se e [FuTa2]) . P R O O F . W e can assum e tha t LQ, LI an d th e topologica l isotop y betwee n the m is supporte d i n a 3-bal l B C S 3. Le t f t: B — • B, t £ [0,1 ] , b e a n ambien t isotop y which move s LQ to L\. W e ca n assum e tha t / i i s a contactomorphis m betwee n tubular neighborhood s UQ D LQ and U\ D L\ o f L 0 an d L\. Th e assumptio n o f the theore m guarantee s tha t th e push-forwar d contac t structur e f = (/i)*(f ) o n B\XJ\ i s nomotopi c a s a plan e fiel d t o f vi a a homotop y fixe d o n th e boundary . Thus accordin g t o the classificatio n o f overtwisted contac t structure s fro m [E4 ] th e overtwisted contac t structure s f an d f o n B \ U\ ar e isotopic relativ e t o th e bound ary. Thu s withou t los s o f generalit y w e ca n assum e tha t f\: (I?,f )— > (-B,f) i s a contactomorphism ove r th e whol e bal l B. Se t f t = (ft)*{£,)• Thu s ft , t € [0,1 ] , i s a loop wit h th e bas e poin t f 0 = f i = f i n th e spac e o f C overtwisted contac t struc tures o n B, fixe d o n th e boundary . Accordin g t o th e sam e classificatio n resul t fo r overtwisted structure s fro m [E4 ] th e inclusio n o f this spac e int o th e correspondin g space T> o f plan e field s o n B i s a homotop y equivalence . I t i s possibl e t o assum e without los s o f generalit y tha t th e loo p ft , £ € [0,1 ] , i s contractibl e i n V. Indeed , if thi s i s not th e case , the n on e ca n compensat e th e homotop y clas s o f this loo p b y implanting int o a smal l bal l BQ C B\Ui an additiona l loo p ft , t £ [1 , 2], which re alizes th e homotop y clas s o f the invers e loop . I n vie w o f the homotop y equivalenc e between th e space s C an d T> w e ca n assum e tha t th e adde d loo p consist s o f con tact structures , an d thu s usin g Gray' s theore m w e can find a n isotop y ht'- B— > B, t E [1 ,2] , supporte d i n B 0 an d suc h tha t hi = I d an d (/i*)*(f ) = f t fo r t G [1 ,2] . Set f t = h to fi for t e [1 ,2] . The n w e hav e f t = (/t)*(f ) fo r al l t £ [0,2] . Notic e 11 i.e. overtwiste d outsid e o f an y compac t set . I t i s prove n i n [E8 ] tha t E 3 , fo r instance , ha s a uniqu e overtwiste d a t infinit y contac t structure .

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50

Y. ELIASHBER G AN D M . FRASE R

that $2 is a contactomorphis m (i?,£ ) —> (-B,£ ) whic h move s LQ to L\. Th e loo p £t, t G [0, 2] is contractibl e i n C. Conside r th e two-parametri c famil y £ t,u> £ £ [0 ? 2], 7x £ [0,1 ] , i n C which realize s thi s contraction , s o tha t £ i?o — €t a n d &, i = £ fo r all £ G [0, 2]. Agai n applyin g Gray' s theorem , w e find a famil y o f diffeomorphism s gt,u suc h tha t {gt,u)*{Z>t) = €t,u for al l t G [0,2], u G [0,1], an d g 0)U = g2,u = I d fo r u G [0,1]. Th e compositio n F t = SU(2 ) = 5 3 ; an d th e degree k o f thi s ma p i s (u p t o sign ) th e secon d Cher n clas s c 2 o f th e SU(2)-bundl e over S 4 (evaluate d o n th e fundamenta l cycl e [S 4]), whic h completel y determine s the bundl e u p t o isomorphism . (Se e [1 9 , p . 66 ] fo r a precis e descriptio n o f thi s 'gluing' o f connections) . (Anti-)self-dua l SU(2)-connection s ove r S 4 ar e usuall y called instantons an d th e degre e k o f th e gaug e transformatio n i s als o calle d th e instanton number. Instanton s wit h k < 0 are ofte n calle d anti-instantons. We us e instanton s wit h k — ±1 t o construc t Sasakia n 3-structure s o n 5 7 , th e total spac e o f th e SU(2)-bundl e ove r S 4 wit h c 2 = =Fl . Fix a basi s { n , T 2,7-3} o f th e Li e algebr a su(2) , wher e

that is , we consider su(2 ) a s subalgebra o f the 2 x 2 comple x matrices. U p to a facto r i th e Ti are just th e Paul i matrices , an d the y satisf y th e commutatio n relation s 3

fe = l

In term s o f thi s basi s w e write th e connectio n form s A an d a a s A = A 1 r1 +A

2T2-\-A3T3

and a = OL\T\ + OL2T2 -h OL3T 3,

respectively. Let ft denot e th e curvatur e for m an d F — a* ft it s pullback . I n term s o f th e local form s A an d F, th e structur e equatio n become s F = dA+

l

-[A,A\.

Writing F = F i n + F 2T2 +

F37 3

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58

H. GEIGE S AN D C . B . THOMA S

and usin g th e commutatio n relation s fo r th e r ^ give n above , w e obtai n Fi =dAi+A 2 AA 3, F2 = dA 2 + A 3AAu F3 = dA 3 + A 1 AA2, or, i n concis e form , Fi = dAi + -^2 tijkAj

A A k.

3,k

We now choose a particular instanton . Followin g Atiyah [1 ] we shall use quater nionic notation , tha t is , w e identif y su(2 ) wit h th e purel y imaginar y quaternion s by settin g n =h

T

2= j , r

3

= k.

Now, accordin g t o Atiya h [1 ] , any (anti-)instanto n wit h instanto n numbe r — 1 can b e writte n (u p t o gaug e equivalence ) a s A(x) = I m

\2{x- b) dx 1 + A 2|x-6|2'

where x £ H = M 4 i s a quaternioni c variable , b is a quaternioni c constant , an d A a positive rea l scalar . W e shal l cal l thi s th e basi c (anti-)instanton . See [1 ] fo r a descriptio n o f thi s solutio n i n th e wide r contex t o f th e Atiyah Drinfeld-Hitchin-Manin (ADHM ) construction . To simplif y notation , w e shall specializ e furthe r t o

^

)=

Im

TTR^

Instantons wit h instanton numbe r + 1 are obtained b y replacing x b y x through out. Henceforth , w e shall only dea l wit h anti-instantons ; th e constructio n fo r (+1 ) instantons work s analogously . We shal l no w prov e tha t th e 1 -form s c*i , a 2, a 3 correspondin g t o thi s loca l connection for m A yiel d a Sasakia n 3-structur e o n th e tota l spac e M = S 7 o f th e SU(2)-bundle ove r S 4 define d b y c 2 = 1 . Th e proo f wil l b e divide d u p int o severa l lemmas. LEMMA 8 . The components Cli of the curvature 2-form Q of the basic instanton are non-degenerate on the horizontal distribution in TM defined by the connection 1-form a. PROOF.

W e hav e ••dA + -[A,A] Im Im =Im

dx Adx

l + M2

+ xd

dx A dx

iTW~ (

1 + XX

x dx A x dx

i + M2)2

x dx A x dx (l + \x\ 2)2 xx dx A dx xdx Ax dx 2 2 + 1 + X |2\2 (i + M )

A dx +

dx Adx 1

(l + |x| 2 ) 2 J-

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C O N T A C T S T R U C T U R E S O N 7-MANIFOLD S 5

9

Write dx = dx\ + dx2 i + dx% j + ^4 ft, then dx Adx = 2[(dx\ A d^2 — dx3 A dx^ji + (d# i A dxs + d#2 A dx4)j + {dx\ A dx4 — dx2 A dx3)fc], so we see that dx A dx is purely imaginar y an d anti-self-dual. Henc e dx Adx =

( l + M 2)2'

Thus alon g a th e components 0 2 o f the curvature ft ar e non-degenerate 2-form s on th e horizontal distributio n determine d b y the connection form , tha t is , for all x G l 4 th e Qi(a(x)) ar e symplectic form s o n the 4-dimensional horizonta l spac e 3

Ha(x) = P | kerai>0.(a.) C T

a{x)M

2=1

at a(x). I f we assume fo r the moment tha t thi s non-degenerac y conditio n hold s everywhere ove r R 4 (i.e. , on any horizontal spac e H y wit h ir(y) G M 4 = 5 4 - {00}) , then w e see that th e components f ^ ar e in fact non-degenerat e o n the horizonta l distribution everywhere ; thi s follow s fro m th e local expressio n fo r F abov e an d the fact tha t compactifyin g R 4 t o 5 4 involve s th e conformal facto r ( 1 + |#| 2 ) 2 i n the quadratic ter m Q. Alternatively , on e can write dow n explicitl y th e potential and curvature aroun d 0 0 and observe tha t the y hav e the same for m a s around 0 ; cf. [1 ] . So i t remain s t o prov e tha t th e Hi are non-degenerate o n the horizontal dis tribution ove r R 4 . Thi s w e shall d o below i n conjunctio n wit h th e definition o f the 4>i. • LEMMA

9 . The a2- are contact forms on M and the Reeb vector fields & are

vertical PROOF.

W e hav e

e*i A (dai)3 = OL\ A (fti - a 2 A a 3 ) 3 = - 3 a i A a 2 A a 3 A Vt\. This i s nowhere zer o becaus e o f the non-degenerac y o f Oi on the horizontal distri bution. Then fro m 0 = i^dai = i^ (Tli - a 2 A a3 ) we ge t i^Cli = 0 , an d the non-degenerac y o f f2 i o n the horizonta l distributio n implies tha t £ 1 is vertical. • Observe tha t i^dcti = 0 implies ai(£j) = 6ij. Now w e shall defin e th e fa an d the metric g tha t wil l giv e u s a Sasakia n 3structure (; , &,a2,0)2=1,2,3 o n M The local construction abov e implies that ove r 5 4 — {00} = R 4 we have a section a suc h tha t cr*fii = j[dx\ A dx2 — dxs A dx^), cr*^2 = f(dx\ A dxs + dx2 A dx±), 0"*^3 = f{dx\ A dx4 — dx2 A dx^), where / i s a strictly positiv e functio n o n R4 . Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

60

H. GEIGE S AN D C . B. THOMAS

Define almos t comple x structure s J\ , J2 , J3 o n M 4 by Jidxi = —dx 2i Jidx

3

=

^ 45

J2dXl = —dX4, J?>d

X2

= dX3,

Then J\ J2 = —J2J1 — J3, an d w e hav e a*ni(JlXo,Xo)>0 ( X

0^0)

and a * ^ ( J 2 X o , JiY 0) = a*n i(X0,Y0) for al l vecto r fields XQ, YQ on M 4. Given a horizontal vecto r X i n TM, base d a t som e poin t cr(x) , defin e X Q = TT*X. The n (iX = (a*JiX 0)h. We mak e th e followin g observations : (i) 7r*iX = 7r*((a*JiXo)h) = 7r*cr*JiXo — JiXo, (ii) fafaX = fa(cr*J2X 0)h = {cr*JiJ2X 0)h = (cr*JsXo)h = faX, (iii) similarl y 1 are not everywhere non-degenerate on the horizontal distribution in TM and related by the quaternions, and consequently the connection form a = ( a i , a 2 , a 3 ) does not define a contact 3-structure.

7. Hypercontac t Structure s In [1 4 ] w e introduce d th e followin g concept . 1 4 . A tripl e o f contac t form s (ai,a2,a3 ) o n a manifol d JV 4n+3 i s called a hypercontact structure i f there exist s a Riemannian metri c g and a compat ible almos t contac t 3-structur e (0i , £2,772)2=1,2,3 such tha t g(X,4>iY) = doti(X,Y) for al l vecto r field s X , Y o n JV 4n+3 . Her e compatibilit y i s understoo d i n th e sens e of Definitio n 2 (2c) wit h a z replace d b y rji. DEFINITION

This notio n generalize s tha t o f a Sasakia n 3-structure . I t stil l preserve s th e essential analogie s wit h hyperkahle r structures , bu t i t possesse s a topologica l flex ibility tha t distinguishe s i t fro m Sasakia n 3-structures . Fo r instance , th e resul t from [1 3 ] can be rephrased a s saying that ever y closed, orientable 3-manifol d admit s a hypercontac t structure , wherea s th e clas s o f 3-Sasakia n 3-manifold s i s ver y re stricted (cf . [1 2] , where a classificatio n o f normal contac t structure s o n 3-manifold s is given, from whic h a classification o f 3-Sasakian 3-manifold s ca n easily be derived). The mai n resul t o f [1 4 ] i s that i t i s possible t o perfor m 0-surger y o n hypercon tact manifolds . Thi s show s th e existenc e o f hypercontac t structure s o n connecte d sums o f copie s o f S 1 x 5 6 (or , mor e generally , S 1 x 5 4 n + 2 ) . The method s o f [1 3 ] als o allo w othe r simpl e application s relevan t t o dimensio n seven. Her e i s a sampl e resul t (cf . [1 4]) . PROPOSITION 1 5 . If M is 2-connected, then it admits three pointwise linearly independent contact forms with pointwise linearly independent Reeb vector fields.

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CONTACT STRUCTURE S O N 7-MANIFOLD S

63

8. Exoti c Contac t Sphere s an d Spac e Form s 3

On S exoti c contac t structure s wer e discovere d b y Bennequi n an d completel y classified b y Eliashber g [6] . Non-standar d contac t structure s o n sphere s o f dimen sion > 5 have bee n constructe d b y Eliashber g [5] . In dimensio n > 5 exoticit y i s detecte d b y a theore m o f Gromov , Eliashberg , Floer an d McDuf f whic h state s tha t i f (W , cu) i s a symplectic manifol d wit h contac t type boundar y equa l t o S 2n~l wit h it s standar d contac t structur e an d (W , LO) doe s not contai n an y symplectic spheres, then W i s diffeomorphic t o a ball B 2n. On e ca n now procee d i n eithe r o f tw o closel y relate d way s t o construc t contac t structure s on S 2 n - 1 , n > 3 , wit h symplecti c fillin g differen t fro m th e standar d B 2n. Whil e Eliashberg plumbe d copie s o f th e cotangen t dis c bundl e o f S n t o produc e a non standard filling , i t i s also possibl e t o exploi t th e topologica l propertie s o f Brieskor n varieties. Th e secon d approac h ha s th e advantag e o f exhibitin g a n explici t contac t form, whic h permit s t o stud y grou p action s o n thes e spaces , a s w e shal l d o i n th e second par t o f this section . Recal l that th e lin k betwee n plumbin g an d th e algebro geometric constructio n i s provide d b y th e theor y o f 0(n)-manifold s ( n odd ) an d knot manifold s ( n even) . Fo r a n excellen t introductio n t o thi s par t o f th e theor y see th e Bourbak i repor t o f Hirzebruc h [1 6] . Let E = S ^ 7 1 a n ( t ) b e th e intersectio n o f th e non-singula r comple x hypersur face give n b y th e equatio n

*o° + • • • + 4n = * in C n + 1 (wit h t a smal l positiv e rea l number ) wit h th e uni t spher e 5 2 n + 1 c C The rea l 1 -for m

n+1

.

a= ^^—{Zjdzj-Zjdzj) 3=0 j

defines (fo r sufficientl y smal l t) a contac t for m o n E ( a o , . . . , a n ), se e [22] . A sym plectic (indee d a holomorphic ) fillin g (W,LJ) i s provide d b y

w = {(^ ^)ec^I n^

+i

1 ^;^=*,x^kii 2 1 , then w e hav e th e followin g resul t of Brieskorn , cf . [1 6] . 1 6 . Let n > 3. Then E^ 71 -1 is a topological sphere if and only if T(a) = r ( a 0 , . . . , a n ) satisfies one of the following conditions: (a) T(a ) has at least two isolated points, or (b) T(a ) has one isolated point and at least one connectedness component K with an odd number of vertices such that gcd(a^ , aj) = 2 for ai, aj G K (i ^ j). PROPOSITION

Specializing t o n = 4 we have th e followin g 7-dimensiona l examples . Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

H. G E I G E S A N D C . B . T H O M A S

64

1. The grap h r(3,6f c — 1 ,2,2,2 ) ha s tw o isolate d points . Henc e £(3,6f c — 1,2,2,2) i s a topologica l sphere , whos e diffeomorphis m typ e i s determine d by the congruence class of k mo d 28. We therefore hav e (for k = 0 mo d 28 ) an infinit e famil y o f Brieskorn varieties , eac h diffeomorphi c t o *S 7, carrying exotic contac t forms . £(6,7 — 1,18j — 1, 3,3,3) ha s the same graph a s before, bu t with all exponents odd. Thi s wil l b e importan t i n th e discussio n o f involution s below . Th e diffeomorphism typ e i s determine d b y th e congruenc e clas s o f 9j(4j — 1 ) mod 28 , se e [1 7 , Formul a (5)] , so agai n w e hav e infinitel y man y model s ofS7. The explici t natur e o f the form furthe r allow s u s to determin e th e homotop y class o f the underlying almos t contac t structure . I n the case o f 5 7 , and probabl y more generally , fo r even values of n, this underlyin g almos t contac t structur e i s not homotopic t o the standard one. For n odd one can give concrete examples where the contact structur e o n S 2n~l is exoti c bu t homotopicall y standard . Fo r instance, £(2,2,2,2,2,49 ) i s diffeomor phic t o S 9 (rathe r tha n th e Kervair e sphere) , an d the induce d contac t structur e is exoti c bu t homotopicall y standard . Moreover , on e can give explici t realization s of contac t structure s o n S 2n~l, n odd , in every homotop y clas s o f almost contac t structures. Next conside r spherica l spac e forms , i.e . quotients o f S 2n~l b y free action s of finite group s G. W e start wit h linea r actions , wher e w e have th e followin g resul t due t o Wolf [30] . 1 7 . If the action of G on 52 n _ 1 is induced by a representation of G in 0(2n), then the space form M{G) = S 2n~l jG inherits a contact structure from the standard structure on S 2n~l. THEOREM

This i s a consequenc e o f th e fac t tha t a discret e subgrou p o f 0(2n ) actin g freely o n S 2n~l i s conjugate i n 0(2n) to a subgroup o f U(n), an d the latter grou p preserves th e standard contac t for m a$ on S 2n~l. Conversely, i t i s interesting t o ask for the restrictions whic h mus t b e impose d on a free, smoot h (bu t non-linear) G-actio n i n order fo r it to preserve th e form LJQ. More generally , ca n we construct contac t spac e form s wit h fundamenta l grou p iso morphic t o G which ar e not included i n Theorem 17? Examples. Dimension 3 . Her e i t i s known that , i f G is solvable an d G preserves ao , then the actio n o f G is conjugate t o a linear action . I f G is not solvable th e situation i s more complicated . Dimension 5 . Th e weak homotopy typ e of the group of strict contac t automor phisms o f (S' 5 ,Q;O) (i.e . those automorphism s preservin g ao , not onl y th e contac t structure kerao ) i s known t o be that o f a compact Li e group. Thi s follow s fro m a result o f Gromov o n the symplectomorphism grou p o f CP 2 an d work o f Banyaga , and impose s sever e restriction s o n compatible G- act ions. Fo r example, th e actio n restricted t o a cycli c subgrou p o f G must b e conjugate t o a linea r action , an d the classifying ma p ip: BG— > BDiff a(S5) mus t facto r throug h B\J(3). Of themselves thes e condition s d o not exclude action s by the metacyclic group s Dsp wit h 3\p— 1; to settle this question mor e work on contact surger y i s needed. W e recall tha t th e significance o f metacyclic group s o f this kin d i s that, althoug h the y Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

C O N T A C T S T R U C T U R E S O N 7-MANIFOLD S

65

admit fixed poin t fre e representation s i n Diff + (S 5 ), the y admi t non e suc h i n U(3). It i s eas y t o writ e dow n a fre e actio n o n th e Brieskor n variet y S / 3 m J t ) fo r suitable value s o f m an d £, see [26], but thi s i s not a homotop y sphere—compar e Proposition 1 6 above. However , E 5 admit s a contac t for m whic h survive s unde r surgery, sinc e w e onl y hav e t o kil l homotop y classe s i n dimensio n 2 i n orde r t o replace E 5 by a sphere. A s for the non-equivariant example s above, the final contact form i s exotic. Detail s o f this an d related construction s wil l appea r elsewher e [11, 15]. Dimension 7 . Thi s seem s to have some similarity t o dimension 3 . In particular the followin g purel y topologica l resul t show s that w e cannot us e metacyclic group s to construc t exoti c spac e form s wit h contac t structure s i n th e wa y outline d fo r dimension 5. 1 8 . If the finite group G acts freely and topologically on S 7, then there is a free linear action of G either on S7 or on S 1 5. THEOREM

SKETCH O F THE PROOF. I f K i s a subgroup o f G , th e cohomologica l perio d of

K divide s tha t o f G, which i s 2, 4, or 8. Thi s exclude s th e metacyclic group s D pq with q | p — 1, except whe n q — 2. Thi s las t cas e does not occur b y an old argument of Milno r [25] , and the constructio n o f a fre e linea r actio n i s now a case-by-cas e application o f representation theory . Ful l detail s wil l appea r elsewhere . • This theore m show s tha t i n lookin g fo r grou p action s compatibl e wit h som e contact structur e o n S 7, w e ar e restricte d t o finite subgroup s o f SO s o r SOi6 Again usin g Brieskor n varietie s w e can show tha t ther e ar e plenty o f them. If eac h o f th e exponent s a ; i n ( a o , . . . , a n ) i s odd , th e involutio n T z = — z of C n + 1 induce s a fixed poin t fre e involutio n o n E j ^ " . 1 ^ (0 ) whic h i s compati ble wit h th e for m a define d above . W e alread y kno w tha t wit h (ao,...,a4 ) = (6j — 1 ,1 8, 7 — 1 ,3,3,3 ) w e ca n produc e (fo r suitabl e j) th e differentia l 7-sphere , and th e contact structur e induce d b y a wil l be exotic, a s will be the induced for m on the orbit spac e £ / T . Th e reference fo r this materia l i s [1 7] , in which on e of the several equivalen t definition s o f the Browder-Livesa y signatur e p of an involutio n is given . However , th e precise valu e o f p(Hj L an ,T) remains , t o th e bes t o f our knowledge, a n open question . Not e tha t th e corresponding proble m i n dimension 3 has bee n solved , an d that p ( E ? 6 _ 118«_1 3 ) ^ ) — 8 j, see [17, Section 6] . Th e numerical problem s i n passing fro m n — 2 to n = 4 do not see m insurmountable , i n contrast t o those fo r a general valu e o f n. But i t is also possible to fall bac k on the Eliashberg plumbin g construction . B y an interestin g us e of the quaternions Hirzebruc h i s able to use plumbing alon g the tree whic h arise s fro m resolvin g th e singularity a t th e origin o f the surfac e {(z0,zl,z2):z30+z61j-1+z128j-1=0} to sho w tha t fo r each 7-dimensiona l sphere , an d each intege r j , there exist s a free , smooth involutio n wit h Browder-Livesa y invarian t equa l t o 8 j. Assumin g tha t on e can sho w tha t th e contac t structur e o n the boundary o f the plumbed manifol d i s compatible wit h th e involution, th e argument abov e ca n be strengthened t o show: The standard sphere S 7 admits infinitely many fixed point free involutions which are pairwise smoothly inequivalent. Each involution preserves a contact form on S 7. THEOREM.

It i s possible t o prove simila r result s fo r the cyclic grou p G p, p an odd prime. Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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H. GEIGE S AN D C . B . THOMA S

References 1. M . F . Atiyah , Geometry of Yang-Mills fields, Scuol a Normal e Pisa , Pis a 1 979 . 2. A . Banyaga , Instantons and hypercontact structures, J . Geom . Phys . 1 9 (1 996) , no . 3 , 267 276. 3. C . P . Boyer , K . Galicki , an d B . M . Mann , Quatemionic reduction and Einstein manifolds, Comm. Anal . Geom . 1 (1 993) , no . 2 , 229-279 . 4. Y . Eliashberg , Topological characterization of Stein manifolds of dimension > 2 , Internat . J . Math. 1 (1 990) , no . 1 , 29-46 . 5. On symplectic manifolds with some contact properties, J . Differentia l Geom . 3 3 (1 991 ) , no. 1 , 233-238 . 6. Contact 3-manifolds twenty years since J. Martinet's work, Ann . Inst . Fourie r (Greno ble) 4 2 (1 992) , no . 1 -2 , 1 65-1 92 . 7. H . Geiges , Contact structures on 1 -connected b-manifolds, Mathematik a 3 8 (1 991 ) , no . 2 , 303-311. 8. Contact structures on (n — l)-connected (2 n + l)-manifolds, Pacifi c J . Math . 1 6 1 (1993), no . 1 , 1 29-1 37 . 9. Symplectic manifolds with disconnected boundary of contact type, Internat . Math . Res . Notices (1 994) , no . 1 , 23-30 . 10. Constructions of contact manifolds, Math . Proc . Cambridg e Philos . Soc . 1 2 1 (1 997) , no. 3 , 455-464 . 11. Applications of contact surgery, Topolog y 3 6 (1 997) , no . 6 , 1 1 93-1 220 . 12. Normal contact structures on 3-manifolds, Tohok u Math . J . (2 ) 4 9 (1 997) , no . 3 , 415-422. 13. H . Geige s an d J . Gonzalo , An application of convex integration to contact geometry, Trans . Amer. Math . Soc . 34 8 (1 996) , no . 6 , 21 39-21 49 . 14. H . Geige s an d C . B . Thomas , Hypercontact manifolds, J . Londo n Math . Soc . (2 ) 5 1 (1 995) , no. 2 , 342-352 . 15. Contact spherical space forms (i n preparation) . 16. F . Hirzebruch , Singularities and exotic spheres, Seminair e Bourbaki , 1 966/67 , Exp . 31 4 , In stitut Henr i Poincare , Paris , 1 967 . 17. Involutionen auf Mannigfaltigkeiten, Proc . Conf . o n Transformatio n Group s (Ne w Or leans 1 967) , Springer , Ne w York , 1 968 , pp . 1 48-1 66 . 18. N . J . Hitchin , The self-duality equations on a Riemann surface, Proc . Londo n Math . Soc . (3 ) 55 (1 987) , no . 1 , 59-1 26 . 19. S . Kobayash i an d M . Nomizu , Foundations of differential geometry I , Interscience , Ne w York London 1 963 . 20. M . Konishi , On manifolds with Sasakian 3-structure over quaternion Kdhler manifolds, Koda i Math. Sem . Rep . 2 6 (1 974/75) , 1 94-200 . 21. R . Lutz , Sur la geometric des structures de contact invariantes, Ann . Inst . Fourie r (Grenoble ) 29 (1 979) , no . 1 , 283-306 . 22. R . Lut z an d C . Meckert , Structures de contact sur certaines spheres exotiques, C . R . Acad . Sci. Pari s Ser . A- B Math . 28 2 (1 976) , no . 1 1 , A591-A593. 23. J . Martinet , Formes de contact sur les varietes de dimension 3 , Proc . Liverpoo l Singularitie s Symposium I I (1 969/70 ) (C . T . C . Wall , ed.) , Lectur e Note s i n Math . vol . 209 , Springer , Berlin, 1 971 , pp. 1 42-1 63 . 24. W . S . Massey , Obstructions to the existence of almost complex structures, Bull . Amer . Math . Soc. 6 7 (1 961 ) , 559-564 . 25. J . Milnor , Groups which act on S n without fixed points, Amer . J . Math . 7 9 (1 957) , 623-630 . 26. T . Petrie , Free metacyclic group actions on homotopy spheres, Ann . o f Math . (2 ) 9 4 (1 971 ) , 108-124. 27. C . B . Thomas , Contact structures on (n — 1)-connected (2n + l)-manifolds, Banac h Cente r Publ. vol . 1 8 , PWN , Warsaw , 1 986 , pp . 255-270 . 28. A . Weinstein , Contact surgery and symplectic handlebodies, Hokkaid o Math . J . 2 0 (1 991 ) , no. 2 , 241 -251 . 29. D . L . Wilkens , Closed (s — 1)-connected (2 s + 1 ) -manifolds, s = 3, 7 , Bull . Londo n Math . Soc . 4 (1 972) , 27-31 . 30. J . A . Wolf , A contact structure for odd-dimensional spherical space forms, Proc . Amer . Math . Soc. 1 9 (1 968) , 1 96 .

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CONTACT STRUCTURE S O N 7-MANIFOLD S 6

7

DEPARTMENT O F MATHEMATICS , E T H - Z E N T R U M , 809 2 ZURICH , SWITZERLAN D

E-mail address: geigesQmath.ethz.c h DEPARTMENT O F P U R E MATHEMATIC S AN D MATHEMATICA L STATISTICS , UNIVERSIT Y O F CAM BRIDGE, 1 6 MIL L LANE , CAMBRIDG E CB 2 ISB , U.K

.

E-mail address: c.b.t homasQdpmms.cam.ac.uk

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https://doi.org/10.1090/crmp/015/04 Centre d e Recherche s Mathematique s CRM Proceeding s an d Lectur e Note s Volume 1 5 , 1998

On th e Flu x Conjecture s Prangois Lalonde , Dus a McDufT , an d Leonid Polterovic h ABSTRACT. Th e "flu x conjecture " fo r symplecti c manifold s state s tha t th e group o f Hamiltonia n difFeomorphism s i s C 1 -closed i n th e grou p o f al l sym plectic diffeomorphisms . W e prove the conjecture fo r spherically rational man ifolds an d for those whos e minima l Cher n numbe r o n 2-spheres eithe r vanishe s or i s large enough . W e also confir m a natura l versio n o f the flux conjectur e fo r symplectic toru s actions . I n som e case s w e can go further an d prov e tha t th e group o f Hamiltonian diffeomorphism s i s C°-closed i n the identity componen t of th e group o f all symplectic diffeomorphisms .

1. Introductio n Let (M,LJ) b e a compac t symplecti c manifol d withou t boundary , an d G — Symp 0 (M, LJ) be the identity component o f the group of symplectic diffeomorphism s of M. Ther e i s an exact sequenc e l

H ->G^H

(M;R)

where G is the universal cove r o f G, H i s the subgroup o f G formed b y those path s which ar e homotopic i n G wit h fixed endpoint s t o a Hamiltonia n path , an d F i s the flux homomorphis m define d b y A t eft. Jo Here th e family o f closed 1 -form s {A^ } generates th e isotopy { te[0)i]} in the sense that F({t})= [

LU

J = At(- ) fo r all t€ [0,1 ] .

Recall als o tha t a pat h i s Hamiltonia n i f it s generatin g famil y {At } is exac t a t each tim e t. Thu s th e exact sequenc e simpl y expresse s th e fact tha t a pat h wit h vanishing (average ) flux ca n be perturbe d t o hav e vanishin g "instantaneou s flux" 1991 Mathematics Subject Classification. 53C1 5 , 58Dxx , 58Fxx . The first autho r wa s partially supporte d b y NSER C gran t O G P 009291 3 an d FCA R gran t ER-1199. The secon d autho r wa s partially supporte d b y NSF grant DM S 9401443. All thre e author s wer e supporte d b y a NSER C Collaborativ e Projec t Gran t CPGO1 63730 . This i s the final for m o f the paper . (c) 1 99 8 American Mathematica l Societ y 69 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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F. L A L O N D E ET AL.

for eac h t. Moreover , i t i s easy to chec k that F({(f) t}) i s the clas s in H l(M', R) tha t assigns t o eac h loo p 7 i n M th e integra l o f u ove r th e cylinde r C 7 : S 1 x [0,1 ]-> M defined b y C 7(z,t) = ^ ( 7 ( 2 ) ) . (Proof s o f th e abov e statement s ca n b e foun d i n Banyaga [1 ] o r McDuff-Salamo n [1 3]. ) W e wil l sometime s refe r t o th e cylinde r C 1 as th e trace o f {(j) t} on 7 . Let u s denot e b y Ham(M,cj) , o r simpl y Ham(M) , th e grou p o f Hamiltonia n diffeomorphisms, tha t i s to sa y thos e element s o f G whic h ar e endpoint s o f Hamil tonian paths 1 . Further , w e define th e flux subgroup V of H l(M\ R) t o b e the imag e by F o f the close d loop s i n G: T - Image(7n(G ) - ^ H\M;R)). If {te[o,i] } is a pat h o f symplecti c diffeomorphism s the n i t i s eas y t o se e tha t it s endpoint i belong s t o Ham(M ) i f an d onl y i f it s flu x F({(j) t}) belong s t o T. I n other words , ther e i s a n exac t sequenc e (*) Ham(M,o;

) - + Symp 0 (M,o;) - • H\M,R)/r,

where the secon d ma p i s induced b y F. Thus , the group T contain s crucia l informa tion abou t th e manifold , bein g th e ke y t o whethe r o r no t a symplecti c diffeomor phism i s Hamiltonian . I n particula r i t i s importan t t o kno w i f T i s discrete , an d i f it is , ho w t o estimat e th e siz e o f a neighborhoo d o f {0 } in ^(M; R) tha t contain s no elemen t o f T excep t {0 } itself. W e begi n wit h th e followin g observation : PROPOSITION 1 .1 . For any closed symplectic manifold, F is discrete if and only if the subgroup of Hamiltonian diffeomorphisms is C 1 -closed in the full group of (symplectic) diffeomorphisms of the manifold.

This follow s almos t immediatel y fro m th e existenc e o f th e exac t sequenc e (* ) since al l the map s involve d ar e C 1 -continuous. A more detaile d proo f i s given i n § 3 below. W e ca n no w stat e th e mai n conjectures : T H E C 1 - F L U X CONJECTURE .

The flux subgroup is discrete for all symplectic

manifolds. T H E C ° - F L U X CONJECTURE . For any symplectic manifold, Ham(M ) is enclosed in the identity component of the group of symplectic diffeomorphisms.

The mai n proble m posed by the flux conjecture 2 ca n be presented i n the follow ing way . Conside r a "long " Hamiltonia n pat h {0te[o,i] } beginnin g a t th e identit y whose grap h a t intermediat e time s escape s ou t o f a Weinstei n neighborhoo d U o f the diagona l i n ( M x M , — u © u ) , bu t whic h come s bac k insid e tha t neighborhoo d at tim e t = 1 . A s i s well-known , U ca n b e identifie d wit h a neighborhoo d o f th e zero sectio n i n th e cotangen t bundl e o f M , and , whe n i t is , a Hamiltonia n isotop y that stay s insid e U i s given b y th e graph s o f a famil y o f exact 1 -forms . However , i f the pat h goe s outside Z7 , then i n genera l ther e i s no obvious reaso n tha t th e 1 -for m corresponding t o it s endpoin t b e exact . 1

When n o specifi c mentio n i s made , al l ou r path s begi n a t th e identity . Not e als o tha t th e group H whic h appear s i n th e abov e exac t sequenc e i s just th e universa l cove r o f Ham(M , u;). 2 Classically, th e "flu x conjecture " i s what w e call her e the C^-flu x conjecture . Fo r thi s reason , when w e refe r t o th e "flu x conjecture " wit h n o qualification , w e alway s hav e th e C 1 -case i n mind .

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ON T H E F L U X C O N J E C T U R E S

71

Note als o tha t th e statemen t tha t Ham(M ) i s C^-close d fo r on e k > 1 i s obviously equivalen t t o th e statemen t tha t i t i s C fc-closed fo r al l k > 1 . However , the questio n o f whethe r i t i s C°-close d i s a priori quit e different , an d seem s t o b e very difficul t t o decid e excep t i n som e particular case s lik e tori (se e below). Thi s i s a fundamental issue . Indee d al l known invariants (suc h as symplectic homology) ar e C°-invariants attache d t o Hamiltonia n paths, bu t on e i s actuall y intereste d i n th e dependence o n th e Hamiltonia n endpoint Observ e als o that , althoug h symplecti c rigidity tell s u s tha t th e grou p Symp(M ) o f al l symplectomorphism s i s C°-close d in th e grou p o f al l diffeomorphism s o f M , i t i s no t know n whethe r o r no t th e identity componen t Symp 0 (M) o f Symp(M ) i s C°-close d i n Symp(M) . T o avoi d this question , w e look her e a t th e C°-closur e o f Ham(M ) i n Symp 0 (M). As will become clear in this paper, th e flux conjectures li e at th e very borderlin e between sof t an d har d symplecti c topology . Wha t w e do her e i s to sho w ho w har d techniques an d thei r consequence s ca n b e use d t o prov e thes e conjecture s i n man y cases wher e purel y soft , topologica l method s see m t o fail . 1.1. Som e ol d results . Th e flu x grou p T wa s firs t explicitl y mentione d i n Banyaga's foundationa l pape r [1 ] , where i t wa s observed tha t T i s discrete whe n M is Kahler , or , mor e generally , whe n M i s Lefschetz. 3 Fo r th e sak e o f completeness , we will briefl y recal l thi s an d othe r earl y result s o n T. i. I t i s easy t o chec k tha t ther e i s a commutativ e diagra m expressin g th e fac t that th e followin g tw o homomorphism s coincid e u p t o a universa l multiplicativ e constant dependin g onl y o n th e dimensio n o f M: 7Ti(G) ^ 7Ti(M

) - > Hi(M;Z ) - ^ > H

2n

1 -\M' )

and TTI(G)

- ^ H\M,R)

u[uj]n

~\ f f ^ - ^ M j R )

.

Here ev i s th e evaluatio n ma p a t a poin t o f M , PD i s the Poincar e dua l an d F i s the flu x homomorphism . Bu t th e firs t ma p ha s discret e image . Thus , i f a manifol d is Lefschetz, th e subgrou p T C H X(M, R ) i s discrete too . Not e als o that i n this cas e the flu x homomorphis m vanishe s o n al l element s o f 7Ti (G) tha t evaluat e triviall y i n the homolog y grou p Hi(M,M). ii. Temporarily , le t G denot e th e grou p o f diffeomorphism s whic h preserv e some give n structur e o n M . The n th e "c-flux " homomorphis m

defined b y assignin g t o th e pai r (0t€[o,i]>7) £ ni(G) x Hi(M;R) th e integratio n o f the clas s c G H2(M) o n th e trac e o f {4>t} o n 7 , vanishe s identicall y i f c is a charac teristic clas s fo r th e structur e preserve d b y G. Thi s wa s prove d b y McDuf f i n [1 1 ] by a n argumen t base d o n Gottlie b theory . ( A more transparen t proo f wil l be give n in [7]. ) Restrictin g t o th e cas e whe n G i s the grou p o f symplecti c diffeomorphism s and c is the firs t Cher n clas s o f the tangen t bundl e o f the symplecti c manifold , w e see that th e flu x vanishe s fo r monoton e manifolds. 4 (Thi s applicatio n t o monoton e manifolds wa s not explicitl y mentione d i n [1 1 ] , but appear s i n Lupton-Oprea [1 0]. ) 3

This mean s tha t H 1 (M,R) - ^ • H2n-l(M;R) i s a n isomorphism . We recal l tha t a monotone manifol d i s a symplecti c manifol d satisfyin g c — KUJ for som e K > 0 . I t i s semi-monotone i f K > 0 . 4

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iii. I t i s also prove d i n [1 1 ] that V is discrete whe n th e clas s o f the symplecti c form i s decomposable, tha t i s to sa y when [uo] i s the su m o f products o f elements of

Hl(M,R).

iv. Finall y i t follow s immediatel y fro m Ginzburg' s result s i n [5 ] that an y sym plectic torus actio n on a Lefschetz manifol d ha s discrete flux: see §2 for more detail . 1.2. Th e evaluatio n map . I n thi s sectio n w e discuss th e relatio n o f the flux conjectures t o th e topologica l propertie s o f th e orbit s {4> t(x)} o f a loo p {4>t}- Our work i s base d o n th e followin g dee p fact : For ever y Hamiltonia n flow { te[0,i]} o n a close d symplecti c man ifold, ther e i s a t leas t on e fixed poin t x £ M o f i suc h tha t th e loop {i wit h contractible orbit s {(j)t(x)}\ se e Hofer-Salamon [6] , Fukaya-Ono [4 ] and Liu-Tian [9]. As w e will se e i n §3 , the followin g resul t i s a n almos t immediat e consequence . PROPOSITION 1 .2 . The flux homomorphism has discrete image on K\ (G) if and only if it has discrete image on the subgroup K of TTI(G) formed by loops with contractible orbits in M.

Thus th e flux conjectur e onl y depend s o n thos e symplecti c loop s which , lik e Hamiltonian ones , hav e contractibl e orbits . On e sometime s say s tha t th e flux ho momorphism o f som e symplecti c manifol d factorizes i f i t factorize s throug h th e evaluation ma p 7Ti(Symp(M) )— » TTI(M). I n othe r words , i t factorize s whe n ever y symplectic loo p wit h contractibl e orbi t ha s vanishin g flux. Propositio n 1 . 2 show s that factorizatio n implie s discreteness . REMARK 1 .3 . Althoug h bot h th e abov e statement s remai n tru e fo r compactl y supported Hamiltonia n flows on noncompact manifolds , the y d o not exten d to flows of arbitrar y support , eve n i f on e consider s th e evaluatio n i n homology. Indeed , th e standard rotatio n of the annulus is Hamiltonian. O n the other hand the fact tha t th e evaluation i n rea l homolog y vanishe s fo r Hamiltonia n loop s o n close d manifold s i s very elementary an d follow s fro m th e commutativ e diagra m i n §1 .1 . Thi s illustrate s a strikin g contras t betwee n th e homologica l (mo d torsion ) an d homotopica l point s of view o n th e evaluatio n map : se e Bialy-Polterovic h [2 ] for furthe r discussion .

In vie w o f th e abov e results , on e migh t wonde r i f an y symplecti c loo p wit h contractible orbit s ha s t o b e Hamiltonia n u p t o homotopy . I n dimensio n 4 th e answer t o thi s questio n i s i n th e affirmative , bu t i t i s fals e i n general . Indeed , on e has th e followin g result : PROPOSITION 1

.4 . Let (M , LU) be a closed symplectic manifold.

(i) In dimension 4 every symplectic S 1 -action with contractible orbits is Hamiltonian (and hence has fixed points). (ii) There is a non-Hamiltonian symplectic S 1 -action on a 4-manifold with orbits that are homologous to zero but not contractible. (iii) In dimension 6 there is a non-Hamiltonian S 1 -action with fixed points and hence with contractible orbits. Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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We wil l see i n § 2 that par t (i ) follow s almos t immediatel y fro m result s i n Mc Duff [1 2] . I t i s eas y t o construc t a n exampl e fo r (ii ) i n whic h M i s a nonKahle r T 2 -bundle ove r T 2 , fo r exampl e th e Kodaira-Thursto n manifold , whil e a n exampl e of typ e (hi ) ma y b e foun d i n [1 2] . The abov e resul t show s tha t i t i s no t i n genera l possibl e t o distinguis h Hamil tonian loop s from other s by looking only at th e topological properties of their orbits . However, th e stor y become s mor e interestin g i f on e look s a t th e "evaluatio n map " ev^: Hk{M) — > Ufc+i(M ) o f a loo p {4>te[o,i]} m dimension s k > 0. Here , ev^ i s th e map whic h takes a fc-cycle 7 t o it s trace C 1 = U t{(f)t(j)}- Thu s { t} in Dif f (M) i s Hamiltonian wit h respec t t o UJ an d i s homotopi c t o a loo p /? ' which i s symplecti c with respec t t o som e othe r symplecti c for m u/ , the n f3' is necessaril y Hamiltonian (up t o homotopy ) wit h respec t t o UJ' . Thi s implie s tha t i f a loo p i s Hamiltonia n with respec t t o u; , then , fo r an y symplecti c for m UJ' sufficientl y clos e t o u; , i t ca n be homotope d t o a ctZ-symplecti c loo p whic h i s actuall y Hamiltonian . Not e tha t this resul t ca n b e interprete d a s a n obstructio n i n th e followin g way . Le t / ? be an y u;-symplectic nonHamiltonia n loo p and denot e b y Ap^ ^ {0 } the imag e of the ma p #1(M, Z) - > H 2{M, Z ) induce d b y (3. Let a e H 2(M,R) b e an y clas s that vanishe s on Ap iUJ. The n a deformation o f the symplecti c structur e fro m th e clas s [UJ] t o a—if it exists—canno t b e lifted t o a deformation o f the loop j3. In other words, each spac e ApiUJ give s a n obstructio n eithe r t o th e deformatio n o f UJ o r t o th e deformatio n o f the imag e o f TTI (Symp(M,u;)) insid e 7i"i(Diff(M)) . Becaus e a symplecti c structur e UJ an d a n u;-symplecti c loo p (3 determin e a symplectic fibratio n V wit h fibe r (M , UJ) over the 2-sphere , thi s can also be interpreted i n the followin g way . Th e existence of a rule d symplecti c for m o n V compatibl e wit h a given symplecti c fibratio n V — > S2 depends onl y o n th e connecte d componen t o f tha t fibratio n i n th e spac e o f al l symplectic fibrations. Thes e result s an d variou s generalization s an d corollarie s wil l appear i n ou r forthcomin g pape r [7] . 1.3. Th e C 1 -flux conjecture . W e no w presen t a lis t o f manifolds fo r whic h we hav e bee n abl e t o confir m th e C 1 -flux conjecture . We star t ou r discussio n wit h th e th e cas e o f symplecti c toru s actions . T o motivate it , suppos e that , fo r som e symplecti c manifol d M , th e grou p G retract s onto a finit e dimensiona l Li e subgrou p H. (Thi s i s know n t o b e tru e fo r som e "simple" symplecti c manifolds, fo r instance for surfaces an d some of their products. ) Then al l elements o f the fundamenta l grou p o f G are represented b y elements o f th e fundamental grou p o f a maxima l toru s insid e i7 , an d th e flu x conjectur e reduce s to th e sam e conjectur e abou t th e fluxe s o f a n autonomou s actio n o f th e torus . Ou r first resul t confirm s th e flu x conjectur e i n thi s case . .5 . LetT n act symplectically on a closed symplectic manifold. Then the restriction of the flux homom,orphism to TTI (Tn) has discrete image in H l(M,R). THEOREM 1

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The proof , whic h i s give n i n §2 , combine s idea s fro m Ginzbur g [5 ] wit h a n analysis o f the Morse-Bot t singularitie s o f the correspondin g (generalized ) momen t map. Let u s g o bac k t o th e genera l non-autonomou s case . Not e tha t th e conjec ture obviousl y hold s i f the integratio n morphis m f u : Hj (M , Z)— > R ha s discret e image, wher e H\\M, Z ) denote s th e se t o f classe s tha t ca n b e represente d b y con tinuous map s o f th e 2-torus . Th e firs t statemen t belo w show s tha t i t i s enoug h t o require tha t thi s integratio n morphis m ha s discret e imag e o n spherical classes # 2 5 (M,Z) = Im(7r 2(M) - > # 2 (Af,Z)) alone. Th e secon d statemen t refer s t o th e minima l spherica l Cher n number , whic h is b y definitio n th e nonnegativ e generato r o f th e imag e o f H^iM^'L) b y th e firs t Chern clas s c — c\(TM) o f th e tangen t bundl e o f M. .6 . The C 1 -flux conjecture holds in the following cases: (i) The manifold is spherically rational, that is to say the image of the spherical 2-classes i7t}- Thu s on e ca n compare th e Floer-Noviko v homolog y o f a generic perturbation o f {ip t} t o the Floe r homology o f th e pat h {fit}- Morally , thi s shoul d lea d t o stron g constraint s o n th e class a. I n man y cases , the y ar e stron g enoug h t o giv e a complet e descriptio n o f T. I f on e i s onl y intereste d i n provin g discreteness , on e ca n limi t thi s stud y t o th e comparison o f thes e tw o path s whe n A a i s s o C 1 -small tha t it s zero s ar e exactl y the fixed point s o f ty\. I n thi s case , on e compare s th e ordinar y Morse-Noviko v homology o f the smal l 1 -for m A G to som e Floe r homology . I n particular , i f A a doe s not hav e enoug h zeroe s t o satisf y th e constraint s o f th e Arnol d conjectur e o r i f th e zeroes d o no t hav e th e appropriat e indices , on e conclude s tha t i t canno t belon g to T . Theorem 1 . 6 extend s th e result s mentione d i n §1 . 1 above . Indeed , a n imme diate corollar y o f (i ) an d (ii ) i s tha t th e flux conjectur e hold s fo r semi-monoton e manifolds. Als o (i ) implie s discreteness i n the decomposabl e case , sinc e any decom posable for m vanishe s o n al l spherica l 2-classes . 1.4. Th e C°-flu x conjecture . W e complet e th e introductio n wit h a discus sion o f the C°-flu x conjecture . Again , se t G = Symp 0 (M), an d le t G{ (i — 0 , 1 ) b e the enclosur e o f Ham(M) i n G. Le t I \ b e the imag e under th e flux homomorphis m of th e lif t o f Gi t o th e universa l cove r o f G . I t i s no t har d t o chec k tha t i n thi s language Propositio n 1 . 1 state s tha t Closure(r ) = Ti , whil e the flux conjecture an d the C°-conjectur e ar e equivalent t o the statement s r = Y\ an d T = T o respectively . It turn s ou t tha t whe n M i s Lefschetz th e grou p T o i s containe d i n a grou p Ytop C H 1 (M , R) whic h depend s onl y on the topology o f M an d o n the cohomolog y Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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class o f the symplecti c form . Namely , conside r th e spac e o f all smoot h (o r eve n continuous) map s M — > M, and denot e b y Map0 (M) th e connecte d componen t of th e identity . Th e flu x homomorphis m F extends naturall y t o a homomorphism 7Ti (Map0 (M))— > H l(M, R ) b y integrating th e symplectic for m o n the evaluation of the path o f maps on the 1 -cycle s of M. Denot e by Ttop its image F (7Ti(Map 0 (M))). Clearly, V C T 0 . THEOREM 1

.7 . If M is Lefschetz, then T 0 C r t o p .

See § 4 fo r th e proo f o f this theorem . Sinc e T C To w e immediatel y ge t th e following usefu l consequence . COROLLARY 1 .8 . Assume that M is Lefschetz and that T t0p = T. Then the C°-flux conjecture holds for M.

As a n example, tak e M to be a close d Kahle r manifol d o f nonpositive cur vature suc h tha t it s fundamenta l grou p ha s n o center . Fo r instanc e a product o f surfaces o f genus greate r tha n 1 has thi s property . I t is easy t o se e that i n this cas e 7Ti(Map0(M)) i s trivial 5 an d henc e T top = {0}. Moreover , M is Lefschet z sinc e it is Kahler. Thu s th e previou s corollar y implie s tha t th e C°-flu x conjectur e hold s o n M. Another immediat e applicatio n o f thi s corollar y i s that th e C°-flu x conjectur e holds fo r th e 2n-dimensiona l toru s wit h a translation invarian t symplecti c struc ture. Indeed , i n this cas e M is Lefschetz s o tha t th e flux homomorphis m factor s through Hi(M )Z). Moreove r translation s generat e a larg e enoug h subgrou p of Sympo(M) fo r u s to se e that 7Ti(Symp 0(M)) map s ont o H\(M, Z) . Thu s T — Ttop, and so , by the abov e theorem, th e conjectur e holds . W e refer th e reade r t o McDuff Salamon [1 3 ] fo r th e explici t computatio n o f T for tori . Another proo f o f the C°-conjectur e fo r thes e tor i wa s first observe d b y Herma n in 1 983 , just afte r Conle y an d Zehnder' s proo f [3 ] of Arnold's conjectur e fo r th e torus. Hi s ide a consist s i n th e followin g basi c observation : i f a path {^t } ha s flux a an d a n endpoin t ip that i s equal t o th e C°-limi t o f Hamiltonian diffeomorphisms , then, b y composing everythin g b y an appropriate Hamiltonian , w e ma y assum e that ip is the translatio n o f flux a. We must sho w that ip is Hamiltonian. But , sinc e all Hamiltonian diffeomorphism s hav e a fixed point b y Arnold' s conjecture , s o does the limi t ip. Henc e ip must b e th e identity , which , o f course , i s Hamiltonian . A natura l generalizatio n o f Herman's ide a i s to replace th e countin g o f fixed points b y a n analysi s o f th e limitin g behavio r o f Floe r homologie s associate d t o a sequence o f Hamiltonian diffeomorphisms , whic h i s jus t th e approac h w e tak e in Theorem 1 .6(ii) . Let us mention finally that th e questions which we discussed abov e can be posed in a more genera l contex t o f Lagrangian submanifolds o f a symplectic manifold . More precisely, le t M be a symplectic manifol d an d L C M be a closed Lagrangia n submanifold. Denot e by H(L) (resp . S(L)) th e space of all Lagrangian submanifold s which ar e obtaine d fro m L by a Hamiltonian (resp . Lagrangian ) isotopy . Th e Clconjecture (i = 0,1) i n this cas e mean s tha t H(L) i s enclosed i n S(L). Se e als o Bialy-Polterovich [2 ] for result s o n th e evaluatio n i n TTI (M, L) i n thi s situation . Throughout th e pape r w e wil l assum e tha t M is a smooth compac t manifol d without boundary . Man y result s appl y whe n M is noncompact, provide d tha t w e 5

Note tha t 7T i (Map0 (Af)) alway s map s int o th e center o f ir\(M).

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consider symplectomorphism s o f compact support . W e leave such extension s t o th e reader. We ar e gratefu l t o M . Braverman , M . Farber , V . Ginzbur g an d J.-C . Sikora v for fruitfu l discussion s o n som e aspect s o f thi s work . 2. Toru s action s We begi n b y provin g Theore m 1 .5 , an d the n discus s th e example s mentione d in Propositio n 1 .4 . 2.1. Th e discretenes s o f T fo r toru s actions . Throughou t thi s sectio n w e use th e identificatio n 7ri(Td) = # i ( T d , Z ) C # i ( T d , R ) . We star t wit h th e followin g usefu l notio n introduce d b y Vikto r Ginzbur g [5] . A symplectic actio n o f T d o n M i s called cohomologically free i f the flu x homomor phism F: Hi(T d,R) — • iJ 1 (M,R) i s injective . Clearly , fo r suc h a n actio n th e flu x subgroup T = F(7Ti(T d)) i s discrete . I t turn s ou t tha t ever y toru s actio n ca n b e reduced t o a cohomologicall y fre e toru s action . Mor e precisely , w e sa y tha t a toru s action o n M i s reducible i f th e followin g condition s (i)-(iii ) hold : (i) ther e exist s a T d -invariant symplecti c submanifol d N C M suc h tha t th e incusion Hi(N,M) — > Hi(M,M) i s onto (w e use th e conventio n tha t a poin t is a symplecti c submanifold) ; (ii) ther e exist s a symplecti c T r -action o n N an d a surjectiv e homomorphis m j : T d— * T r suc h tha t g\^ an d j(g) coincid e a s diffeomorphisms o f N fo r al l

gerd; (hi) r + di m N < d + di m M. This definitio n split s i n tw o cases : eithe r di mTV < d i m M an d i n thi s cas e on e can choos e j a s the identity , o r di mTV = di m M whic h mean s tha t th e whol e actio n factorizes throug h a smalle r torus . As an example, the product o f two standard circl e actions on T 2 x S2 i s reducible since w e may tak e T r = S 1 x i d an d N t o b e th e produc t o f T 2 wit h a fixed poin t onS2. PROPOSITION 2.1 . Every irreducible torus action is cohomologically free. Assum e tha t th e actio n o f T d o n M i s not cohomologicall y free . The n for som e non-zer o elemen t v i n th e Li e algebr a o f T d th e actio n generate d b y v i s Hamiltonian. Denot e b y H th e Hamiltonia n function . Le t L C T d b e th e subtoru s (of positive rank! ) define d a s the closur e o f the 1 -parametri c subgroup V generate d by v. Notic e tha t ever y critica l poin t o f H i s fixed b y th e actio n o f L , sinc e V i s dense i n L. Sinc e th e actio n o f L i s linearizable nea r eac h fixed point , w e conclud e that i n som e loca l comple x coordinate s (zi, ..., z n) o n M nea r a critica l poin t on e can writ e H (u p t o a n additiv e constant ) a s a linea r combinatio n o f th e |z^| 2. I n particular, eithe r H i s identically constant o r H i s Morse-Bott function with even indices. In th e first case , th e actio n o f L o n M i s trivial. Takin g N = M,T r = T d/L, and definin g j a s th e natura l projectio n w e find tha t th e actio n i s reducible . In th e secon d case , tak e N t o b e th e minimu m se t o f H. I t follow s fro m th e previous discussio n tha t N i s a symplecti c submanifold , an d Morse-Bot t theor y implies tha t th e inlusio n Hi(N,M) — > Hi(M,M) i s a n epimorphism . Moreover , PROOF.

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since th e actio n o f L fixe s N an d commute s wit h th e actio n o f th e whol e grou p Td, w e conclude tha t N i s T d -invariant. Therefore , takin g T r — T d an d definin g j as th e identit y ma p w e se e tha t th e actio n o f T d i s reducible . Thi s complete s th e proof. • 1 .5 . Conside r a T d- actio n o n M. Applyin g Proposi tion 2. 1 repeatedl y w e end u p wit h a cohomologicall y fre e reductio n sinc e th e pro cess terminates i n view of (iii) ! I n other words there exists a T rf-invariant symplecti c submanifold N C M an d a homomorphis m j : Td— > T r whic h satisf y condition s (i)-(iii) abov e an d suc h tha t th e T r- actio n o nT V i s cohomologicall y free . Let $ : H^T^R) * H X(N,R) an d F: #i(T d ,]R) - * H^M.R) b e th e flu x homomorphisms o f th e T r -action o n N an d th e T d -action o n M respectively . Le t j * : # i ( T d , R ) - • # i ( T r , R ) an d i* : H^M.R) - > H X{N,R) b e th e natura l maps . Clearly, $ o j , = i * o F, an d thu s PROOF O F THEORE M

i*(F(*1{Td)))=*(j.{n1(Td))). The righ t han d sid e is discrete sinc e j * map s intege r homolog y t o intege r homolog y and $ i s a monomorphis m a s th e flu x o f a cohomologicall y fre e action . Moreover , i* i s a monomorphis m i n vie w o f th e conditio n (i) . Henc e th e flu x subgrou p T = F(7Ti(Td)) i s discrete . Thi s complete s th e proof . • COROLLARY 2.2 . Every symplectic torus action on a closed manifold splits into a product of a Hamiltonian action and a cohomologically free action. P R O O F . Thi s was proved b y Ginzburg [5 ] for th e cas e when either th e symplec tic manifol d i s Lefschetz, o r th e symplecti c for m represent s a n intege r cohomolog y class. Combinin g Ginzburg' s argumen t wit h ou r Theore m 1 . 5 on e immediately get s this statemen t fo r genera l manifolds . •

2.2. S 1 actions . I t wa s shown i n [1 2 ] that a symplectic S^-actio n o n a close d 4-manifold (M , ui) i s Hamiltonia n i f an d onl y i f i t ha s fixe d points . Therefore , part (i ) o f Propositio n 1 . 4 follow s fro m th e nex t lemma . 2.3 . In the 4- dimensional case every symplectic S tractible orbits has fixed points. LEMMA

1

-action with con-

P R O O F . Conside r a n actio n wit h n o fixe d points . Thi s i s generate d b y a non vanishing vecto r fiel d X. A s i n [1 2] , on e ca n slightl y pertur b uo t o a n invarian t form whic h represent s a rationa l cohomolog y clas s an d the n rescal e UJ, t o reduc e to th e cas e whe n th e flu x o f th e action , whic h i s represente d b y th e for m ix(w), is integral . I n thi s cas e ther e i s a ma p / i : M— > S 1 = M/Z , calle d th e generalise d moment map , suc h tha t i x M = fJL*(dt).

Since X ha s n o fixe d point s \i i s a fibratio n wit h fibe r Q an d w e ma y thin k o f M as mad e fro m Q x [0,1 ] b y identifyin g Q x {0 } with Q x {1 } . Moreover , th e actio n of S 1 o n Q x [0,1 ] i s Hamiltonia n an d s o ha s th e usua l structur e o f suc h actions . Thus ther e i s a Seifer t fibratio n n : Q — » E an d a famil y a t o f are a form s o n E suc h that v\Qx{t} =7r*(a t ), [a e [0,1 ] , t] - [a 3] = -{t - s)c, s,t where c G iJ2 (E,Q) i s the Eule r clas s o f n. Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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. L A L O N D E ET AL.

We clai m tha t i t i s impossibl e fo r th e orbit s t o b e contractibl e i n M . T o se e this, observ e tha t th e lon g exact homotop y sequenc e fo r Q — > M — » S 1 implie s tha t 7Ti(Q) inject s int o TTI(M). Henc e i f th e orbit s ar e contractibl e i n M , the y contrac t in Q. 6 Bu t i f the y contrac t i n Q , th e Eule r clas s c must b e nontrivial . Therefor e [cro] 7^ [ Q x {1 } since suc h a ma p woul d induc e a symplectomorphis m (E , CTQ)—> (£ , (7i). • 3. Proo f o f th e mai n theore m i n th e C 1 -case We wil l begi n b y provin g th e elementar y result s state d i n Proposition s 1 . 1 and 1 .2 . W e then prov e th e variou s part s o f Theore m 1 .6 , modul o a lemm a neede d for (ii ) tha t i s presented i n §3.2 . 3.1. Th e mai n arguments . 1 .1 . W e mus t sho w tha t th e C 1 closur e o f Ham(M ) is equivalent t o th e discretenes s o f th e flux grou p T . Let (/>k b e a sequenc e o f Hamiltonia n diffeomorphism s whic h C 1 -converges t o some symplecti c diffeomorphis m (f>. B y composin g th e sequenc e an d it s limi t wit h 4>^1 for som e fixed fco, we ge t a ne w sequenc e ^ o f Hamiltonia n diffeomorphism s converging t o som e ip which i s Hamiltonia n if f 0 is . B y choosin g ko larg e enough , the ne w sequenc e i s C 1 -close t o th e identit y fo r k > &o , an d therefor e ther e ar e closed 1 -form s A ,A & on M suc h tha t th e autonomou s symplecti c pat h V^foii ? iPtelo H ge n e r a t e d b y th e cj-dual s o f A , Xk ar e smal l path s whos e endpoint s ar e V> ipk a n d whos e fluxes ar e equa l t o [A] , [A&] . Fo r eac h k > /co , w e thu s ge t a loo p by composin g (i n th e sens e o f paths, no t pointwise ) a Hamiltonia n pat h {$} fro m the identit y t o ^ = ip k wit h th e pat h { ^ I - * } - Thi s loo p ha s flux — [A&], which i s arbitrarily smal l for k > ko. I f T is discrete, these classes must vanish , an d therefor e their limi t [A ] = 0 . Bu t thi s mean s tha t th e pat h ^ G r 0 ^ i s Hamiltonian . Henc e ip* — i\) i s Hamiltonian, a s required . Conversely, if T is not discrete , there is an arbitrarily smal l class [A ] G H1 (M ; R) which is not i n T but i s the limit o f classes in T. The n the above construction gives a sequence o f Hamiltonian diffeomorphism s whic h C 1 -converge t o a non-Hamiltonia n one. • P R O O F O F PROPOSITIO N

P R O O F O F PROPOSITIO N 1 .2 . W e mus t sho w that , i f the flux homomorphis m F ha s discret e imag e o n th e subgrou p forme d b y symplecti c loop s wit h trivia l evaluation i n 7Ti(M) , it s whol e imag e T i s discrete. T o d o this , i t i s clearl y enoug h to sho w tha t ther e exist s a neighborhoo d U o f 0 i n H 1 (M , R) wit h th e followin g property: Every clas s a G T n U is the imag e unde r F o f a symplectic loo p wit h contractible orbits . If U is sufficiently small , the n ever y clas s a G U can b e represented b y a 1 -for m A which generate s a n autonomou s locally-Hamiltonia n flow ^ whos e onl y close d orbits i n th e tim e interva l t G [0; 1] ar e th e zero s o f A . W e clai m tha t suc h U hav e the require d property . To see this suppose that a G P . The n tpi i s the time-1 map of some Hamiltonia n flow { t} b e suc h a loop , and le t a G H1 (M , R) b e its flux. The n th e value of a on each closed curv e 7 C M i s equal t o the symplecti c are a of the toru s C 1 . Bu t thi s torus i s the imag e of a spher e since on e o f it s generatin g loop s i s contractibl e i n M . Thu s th e homomorphis m

#i(M,R) - ^ H 2(M,R) -^R factors throug h sphere s a: # i ( M , R ) — ^ #

5 2

( M , R ) ±-+ R .

But i f th e manifol d i s sphericall y rational , thi s ma p take s value s i n som e discret e subgroup o f R whic h doe s no t depen d o n a particula r choic e o f a. Thi s implie s th e discreteness o f the flu x subgrou p T. • P R O O F O F T H E O R E M 1 .6(h) . W e mus t sho w tha t i f th e minima l Cher n num ber o f (M , LJ) is zer o o r i s > 2n , the n th e flu x conjectur e holds . Thi s i s base d o n the compariso n o f th e Maslo v an d Floe r indices . In view of Proposition 1 . 2 i t i s enough t o restric t ou r attentio n t o the subgrou p K o f 7Ti(G ) consistin g o f loop s whos e evaluatio n i n TT\ vanishes . Fi x a Riemannia n metric g o n M , an d defin e a nor m ||a| | o f a clas s a G if1(M, R) a s th e C^-g-nor m of the uniqu e ^-harmoni c for m i n this class . Th e followin g lemm a i s proved i n §3.2. LEMMA 3.1 . There exists e > 0 such that every nonzero cohomology class a with \\a\\ < e is represented by a Morse form A which has no critical point of index 0 and 2n, and whose Hamiltonian flow has no nonconstant 1 -periodic orbit.

Granted this , assum e tha t th e imag e o f F(K) i s no t discrete , an d choose e as in Lemm a 3.1 . The n ther e exist s (3 G K wit h 0 < ||-F(/?)| | < e . Represen t th e class F(/3) b y a for m A as i n Lemm a 3. 1 an d denot e b y ^ 1 th e time- 1 ma p o f th e corresponding symplecti c flo w {?/^} . I n vie w o f ou r choice , th e onl y 1 -periodi c orbits o f {ipt} ar e th e constan t ones . Since ipi i s a Hamiltonia n diffeomorphism , w e ca n joi n i t t o th e identit y b y a Hamiltonian pat h {0*} . Introduc e th e followin g notations . Fo r a fixed poin t x o f 0i = ^ 1 denote by i\(x) th e Conley-Zehnder inde x coming from th e flow of A and b y in(x) th e Conley-Zehnde r inde x coming from th e Hamiltonian path . I f the minima l spherical Cher n numbe rA T is 0 o r i s > n , the n Hofer-Salamo n prove d i n [6 ] tha t the Floe r homolog y o f a Hamiltonia n pat h i s isomorphic t o th e ordinar y homolog y with coefficient s i n th e Noviko v rin g o f th e grou p i7< f (M, Z). I n particula r ther e are fixed point s x , y o f 0 wit h iH(x) - i H{y) = 2n (mo d 27V ) (Recall tha t al l relativ e indice s o f Floe r homolog y ar e define d onl y modul o 2N). On th e othe r han d th e orbit s o f th e loo p whic h i s th e compositio n o f {4> t} with Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

80

F. LALOND E ET AL.

{il>i-t} n a v e indice s whic h ar e independen t o f th e choic e o f th e orbit , denot e the m by th e constan t m G Z modul o 2AT . Thus , fo r an y fixe d poin t z G M, i>H(z) — i\( z) —

m

(mo

d 27V )

and therefor e IH(X)

~ iH(y) = h{x) - i\(y) (mo

d 27V) ,

which implie s tha t ix(x) - i x{y) = 2n (mo d 2N). But ou r choic e o f A implie s tha t - 2 n + 2 < i x(x) - i\(y) < 2 n - 2 which mean s tha t 2 < 2 n — (i\(x) - i\(y)) < 4 n — 2 . Th e las t equatio n state s that 2 n — (i\(x) — i\{y)) i s a multipl e o f 27V . Bu t thi s i s impossibl e i fT V is 0 o r > 2rc . D P R O O F O F THEORE M 1 .6(iii) . Suppose , b y contradiction , tha t th e flu x i s no t discrete whe n TT\ acts triviall y o n7T 2 and M ha s dimensio n n = 4 . Becaus e i t i s non-discrete, i t i s no t factorizable . Thu s ther e i s a loo p wit h trivia l evaluatio n i n 7Ti(M) an d non-zer o flux a G H l(M). B y th e commutativit y o f th e diagra m i n §l.l(i), w e must hav e a U [o;n-1] = 0 . Henc e ther e i s a no n zer o clas s suc h that : (i) a U [UJ] = 0 ; an d (ii) ther e i s a loo p o f symplecti c diffeomorphism s {*£[o,i] } whose flux i s a an d which ha s contractibl e evaluatio n i n TTI(M). Now choos e a n elemen t 7 G TT\M such tha t a(7 ) = K ^ 0 , an d le t C — C 7 , th e trace o f {(f) t} on 7 . Then , i f p: M — > M i s the universa l cover , C lift s t o a 2-spher e C i n M , an d

p*(H)(C) = [o;](C)=«^0 . Therefore i t suffice s t o prov e th e followin g lemma. 7 LEMMA

3.2 . Let a be a nonzero class in i7 1 (M;R) such that for some k

aUH f c =0, p*(M*)^0 . Then TC\ acts nontrivially on H2k(M). Moreover, if p*([uj] k) does not vanish on some element of H^M), 7Ti(M ) acts nontrivially on 7T2k{M). P R O O F . Choos e a n elemen t 7 G -K\M such tha t 0(7 ) = K / 0 , an d le t r b e th e deck transformatio n associate d t o 7 . Then , i f th e 1 -for m A represents a , w e hav e p*(X) = dH wher e T*H - H = K. Further ,

A A uo k = dO for som e 2/c-for m 0 on M becaus e aU[cv] k — 0. Thi s implie s d(Jfp*o;fc) = d(p*fl) , which implie s i n tur n tha t Hp*ujk = p*0 + a for som e close d 2/c-for m c r on M. Hence : r-a-a^ (

r *i^

- H)p*w k = np*u k,

7

This lemm a give s som e result s i n al l dimensions . However , i t i s onl y i n dimensio n 4 tha t one get s a clea n statemen t abou t th e flux conjecture .

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ON TH E FLU X CONJECTURE S

81

and s o there i s a 2A:-cycl e C i n M suc h tha t f

a

Jr(C)

-[

a= f

JO

( r V - a) = K [ p*uu k ^ 0 .

JC

JC

Therefore r{C) an d C ar e no t homologou s i n M , an d th e actio n o f TTI(M) o n H2k(M) i s nontrivial . Finally , i f p*([u] k) doe s no t vanis h o n som e elemen t o f j££ffc(M), th e same argument show s that th e action of TTI (M) o n 7T2/c(M) is nontrivial too (not e tha t thi s actio n give n b y the dec k transformation s i s well-defined becaus e 7Ti(M) vanishes) . Becaus e 7T2k(M) i s isomorphic t o 7T2fc(M) , we conclude tha t TTIM acts nontriviall y o n K2k{M). D 3.2. Proo f o f Lemm a 3.1 . I t wa s show n b y G . Levit t [8 ] tha t ever y non zero cohomology clas s can b e represented b y a Morse 1 -for m without critica l point s of inde x 0 an d 2n . W e sho w her e tha t th e Lemm a 3. 1 follow s easil y fro m thi s statement. Howeve r on e ca n als o us e a differen t approac h base d o n th e fac t tha t a generic harmoni c 1 -for m i s Mors e (se e the Appendix) . Here i s a shor t proo f o f Lemm a 3. 1 base d o n Levitt' s theorem . O n som e smal l closed conve x neighbourhoo d V o f 0 G Hl(M,W) choos e an y smoot h sectio n a o f Z1(M) - • iJ^M,!* ) (wher e Z 1 (M) i s th e spac e o f close d 1 -forms ) whic h i s 0 a t 0 (on e ca n d o thi s b y choosin g a Riemannia n metri c say) . The n tak e an y smoot h convex hyperspher e S i n H X(M, M) containing 0 i n it s interior , an d fo r eac h clas s [A] on 5 , choos e a Levit t for m A (which mean s her e a Mors e for m wit h n o critica l point o f inde x 0 o r 2n) an d defin e th e neighbourhoo d U\ C S a s th e intersectio n of S wit h th e classe s [A ] + rV , wher e r i s a smal l rea l number . Eac h elemen t i n U\ is represented b y th e for m A + ra(v). Thu s i f r i s small enough , al l element s o f U\ are represente d b y Levitt form s an d thi s representatio n i s continuous. No w there i s a finite coverin g o f S b y suc h subsets . Thi s implie s tha t ther e i s e smal l enoug h s o that th e hyperspher e eS a s wel l a s it s interio r i s covere d b y Levit t form s tha t ar e such tha t th e Hamiltonia n flow ha s n o non-constan t 1 -periodi c orbit . • 4. Th e C°-conjectur e an d Lefschet z manifold s In thi s sectio n w e prov e Theore m 1 .7 . W e star t wit h a discussio n o f th e prop erties o f th e flux homomorphis m i n th e mor e genera l contex t o f path s o f maps. Let M b e a close d manifold , an d le t a = {t}te[0\i] b e a pat h o f smoot h maps M — > M suc h tha t Q = i d an d \ i s a diffeomorphism . Le t I* b e th e ring o f 0i-invarian t differentia l form s o n M o f degre e k. Defin e th e flux ma p $:/£—» H k~1 (M, R ) a s follows . Give n a n invarian t for m a E I k an d a (k — 1 ) cycle 7 o n M , th e valu e 3>(