Reidemeister torsion in generalized Morse theory

Transcription

1 Foum Math. 14 (2002), 209±244 Foum Mathematicum ( de Guyte 2002 Reidemeiste tosion in genealized Mose theoy Michael Hutchings (Communicated by Andew Ranicki) Abstact. In two pevious papes with Yi-Jen Lee, we de ned and computed a notion of Reidemeiste tosion fo the Mose theoy of closed 1-foms on a nite dimensional manifold. The pesent pape gives an a pioi poof that this Mose theoy invaiant is a topological invaiant. It is hoped that this will povide a model fo possible genealizations to Floe theoy Mathematics Subject Classi cation: 57R70, 57Q10; 37C27, 57R58. In two papes with Yi-Jen Lee [HL1, HL2], we de ned a notion of Reidemeiste tosion fo the Mose theoy of closed 1-foms on a nite dimensional manifold. We conside the ow dual to the 1-fom via an auxiliay metic. Ou invaiant, which we call I, multiplies the algebaic Reidemeiste tosion of the Novikov complex, which counts ow lines between citical points, by a zeta function which counts closed obits of the ow. Fo a closed 1-fom in a eal multiple of an integal cohomology class, i.e. d of a cicle-valued function, we poved in the above papes that I equals a fom of topological Reidemeiste tosion due to Tuaev. This implies a posteioi that I is invaiant unde homotopy of the cicle-valued function and the auxiliay metic. In this pape we epove these esults using an opposite appoach: we st pove a pioi that I is a topological invaiant, depending only on the cohomology class of the closed 1-fom. We then deduce that I agees with Tuaev tosion, by using invaiance to educe to the easie case of an exact 1-fom. This appoach has two advantages. Fist, it woks fo closed 1-foms in an abitay cohomology class, thus extending the esults of ou pevious papes. Second, and pehaps moe impotantly, the poof of invaiance hee should povide a model fo the possible constuction of tosion in Floe theoy. The contents of this pape ae as follows. In 1 we eview the de nition of the invaiant I, state the main esults, and outline the poofs. In 2 and 3 we pove that I is invaiant. The stategy is to study how I vaies in a geneic one paamete family of 1-foms and metics. In 2, we pepae fo this analysis by classifying the bifucations that geneically occu, and we also deal with the complication that in nitely many bifucations may occu in a nite time. The heat of the pape is in 3, whee we analyze what happens in each individual bifucation. While the tosion of the Novikov

2 210 M. Hutchings complex and the zeta function can change, we will see that thei poduct I does not. In 4 we use topological invaiance to give a quick poof that I agees with Tuaev tosion. In 5 we discuss open questions and possible genealizations. Appendix A eviews algebaic aspects of Reidemeiste tosion that ae used thoughout the pape. Appendix B eviews how to emove a cetain ambiguity in Reidemeiste tosion using Tuaev's Eule stuctues. 1 The invaiant I We begin by eviewing the de nition of the invaiant I fom [HL2]. We will emphasize geometic aspects which ae impotant fo the pesent pape, and we genealize [HL2] slightly by allowing di eent abelian coves in Choice 1.2. Afte de ning I, we will state the main esults and outline the poofs. 1.1 Setup and geometic de nitions Let X be a smooth, nite dimensional, closed (connected) manifold with w X ˆ0. We conside a closed 1-fom a and a Riemannian metic g on X. Let V :ˆ g 1 a denote the vecto eld dual to a via g. We wish to count closed obits and ow lines of V, which ae de ned as follows. A closed obit is a nonconstant map g : S 1! X such that g 0 t ˆ lv g t fo some l > 0. Thee is a minus sign because we wok with the ``downwad'' ow as in classical Mose theoy. We conside two closed obits to be equivalent if they di e by a otation of S 1. The peiod p g is the lagest intege k such that g factos though a k-fold coveing S 1! S 1. Fo counting puposes, we can attach a sign to a geneic closed obit as follows. Fo q A g S 1, let U H X be a hypesuface intesecting g tansvesely nea q, and let f q : U! U be the etun map (de ned nea q) which follows the ow V a total of p g times aound the image g S 1. The lineaized etun map induces a map df q : T q X=T q g S 1!T q X=T q g S 1 which does not depend on U. The eigenvalues of df q do not depend on q. We say that g is nondegeneate if 1 df q is invetible. In this case we de ne the Lefschetz sign 1 m g to be the sign of det 1 df q. A citical point is a zeo of a. A citical point p A X is nondegeneate if the gaph of a in the cotangent bundle T X intesects the zeo section tansvesely at p. In this case the deivative `V : T p X! T p X is invetible and self-adjoint; the index of p, denoted by ind p, is the numbe of negative eigenvalues of `V. The descending manifold D p is the set of all x A X such that the tajectoy of the ow V stating at x conveges to p. Similaly, the ascending manifold A p is the set of all x A X fom which the tajectoy of V conveges to p. Ifp is nondegeneate, then D p and A p ae embedded open balls of dimension ind p and dim X ind p, espectively.

3 Reidemeiste tosion 211 If p and q ae citical points, a ow line (of V) fom p to q is a map f : R! X such that f 0 t ˆ V f t and lim t! y f t ˆp and lim t! y f t ˆq. We conside two ow lines to be equivalent if they di e by a tanslation of R. The space of ow lines fom p to q is natually identi ed with D p X A q =R, whee R acts by the ow. Thus, if p and q ae nondegeneate, the expected dimension of the space of ow lines fom p to q is ind p ind q 1. We will impose the following tansvesality conditions. De nition 1.1. The pai a; g is admissible if: (a) All citical points of V ae nondegeneate. (b) The ascending and descending manifolds of citical points of V intesect tansvesely. (c) All closed obits of V ae nondegeneate. A staightfowad tansvesality calculation (cf. [AB, Sc, H]) shows that fo a xed cohomology class aš A H 1 X; R, these conditions hold fo geneic pais a; g. 1.2 Coveings and Novikov ings In the Mose theoy of nonexact closed 1-foms, thee may be in nitely many closed obits and ow lines between citical points of index di eence one. To enable nite counting, we conside a coveing of X. Choice 1.2. We choose a connected abelian coveing p : ~X! X such that p a is exact. Let H denote the goup of coveing tansfomations. Thee is a sujection H 1 X! H, whose kenel consists of homology classes of loops in X that lift to ~X. Ou counting will take place in the Novikov ing L :ˆ Nov H; aš. The meaning of this notation is that if G is an abelian goup and N : G! R is a homomophism, then Nov G; N consists of fomal sums P g A G a g g, with a g A Z, such that fo each R A R, thee ae only nitely many nonzeo coe½cients a g with N g < R. This ing has the obvious addition, and the convolution poduct [N, HS]. Thee is a natual inclusion of the goup ing Z GŠ into the Novikov ing Nov G; N, which is the identity if and only if N 1 0. Example 1.3. Suppose a ˆ df, whee f : X! S 1 is not nullhomotopic. The simplest choice is to take the coveing ~X to be a component of the be poduct of X and R ove S 1. Then H F Z, and the Novikov ing is L F Z t ˆ f P y kˆm a kt k jm; a k A Zg, the ing of intege Lauent seies. This is essentially the setup of [HL1]. (In [HL1], ~X was the entie be poduct of X and R ove S 1. As a esult, t hee equals t N in that pape, whee N is the numbe of components of the be poduct, o equivalently the divisibility of aš A H 1 X; Z.) Fo moe e ned invaiants, one can take ~X to be the univesal abelian cove, as in [HL2].

4 212 M. Hutchings 1.3 Counting closed obits If a; g is admissible, we count closed obits using the zeta function 1:1 z :ˆ exp P g A O 1 m g p g gš A L: (Cf. [F1, Pa3].) Hee O denotes the set of closed obits, and gš A H denotes the image of the homology class g S 1 Š unde the pojection H 1 X!H. Let us eview why z is a well de ned element of L, as the ideas in this agument will be impotant late. Fist, we claim that fo each R A R, thee ae only nitely many closed obits g with aš g < R. The length of such an obit away fom the citical points is bounded above by some multiple of R. An elementay compactness agument [H, Sa1] then shows that an in nite sequence of such obits would accumulate to eithe (i) a degeneate closed obit, o (ii) a ``boken'' closed obit with stopoves at one o moe citical points. Situation (i) would violate admissibility condition (c). In situation (ii), thee would necessaily be a ow line fom a citical point of index i to a citical point of index V i. This would violate admissibility condition (b), since the expected dimension of the space of such ow lines is negative. Let L denote the set of sums P h A H a h h A L such that a h ˆ 0 wheneve aš h U 0. Let L Q :ˆ L n Q. The above paagaph shows that P 1 m g g A O p g gš A L Q : Now exp : L Q! 1 L Q is well de ned by the usual powe seies. Theefoe z A L n Q. But in fact z has intege coe½cients, because thee is a poduct fomula 1:2 z ˆ Q 1 G gš G1 : g A I Hee I denotes the set of ieducible (peiod 1) closed obits, and the two signs associated to each ieducible obit ae detemined by the eigenvalues of the etun map. The fomula is poved by taking the logaithm of both sides, cf. [F2, HL1, IP, Sm]. Remak 1.4. The invese of exp above is also well de ned via the usual powe seies 1 log 1 x ˆPy k 1 x k kˆ1. We will use this fact in 3.4. k 1.4 Counting ow lines We count ow lines using the Novikov complex CN ; q, which is de ned as follows. Let ~ C i denote the set of index i citical points of p V in ~X. Choose f : ~X! R with

5 Reidemeiste tosion 213 df ˆ p a. We de ne CN i to be the set of fomal sums P ~p A ~C i a ~p ~p with a ~p A Z, such that fo each R A R, thee ae only nitely many nonzeo coe½cients a ~p with f ~p > R. The action of H on ~C i by coveing tansfomations makes CN i into a module ove the Novikov ing L. This module is fee; one can obtain a basis by choosing a lift of each citical point in X to ~X. We de ne the bounday opeato q : CN i! CN i 1 by 1:3 q~p :ˆ P hq~p; ~qi ~q ~q A ~C i 1 fo ~p A ~C i. Hee hq~p; ~qi denotes the signed numbe of ow lines of p V fom ~p to ~q. The signs ae detemined as follows [Sa2]. We choose oientations of the descending manifolds of the citical points in X, and lift them to oient the descending manifolds in ~X. Given a nondegeneate ow line fom ~p to ~q, let v A T ~p D ~p be an outwad tangent vecto of the ow line. The ow nea the ow line detemines an isomophism T ~p D ~p =v! T ~q D ~q. We declae the ow line to have positive sign if the oientations on T ~p D ~p and Rv l T ~q D ~q agee. (We do not need to assume that X is oiented.) A compactness agument as in 1.3 and [Sa1, Po, H], using the fact that p a is exact, shows that q is well de ned if a; g is admissible. A standad agument [Po, Sc] then shows that q 2 ˆ 0. The homology of the Novikov complex depends only on the cohomology class aš and the coveing ~X. To descibe it topologically, choose a smooth tiangulation of X, and lift the simplices to obtain an equivaiant tiangulation of ~X. We denote the coesponding chain complex by C ~X ; this is a module ove the goup ing Z H Š. Thee is then a natual isomophism 1:4 H i CN F H i C ~X n Z H Š L : This was stated by Novikov [N], and poofs may be found in [Pa1, Po, HL1]. (Any cell decomposition would su½ce hee, but we will shotly need to estict to tiangulations, in ode to de ne Reidemeiste tosion e ned by an Eule stuctue.) 1.5 Reidemeiste tosion and the invaiant I The Novikov homology (1.4) often vanishes, at least afte tensoing with a eld, and it is then inteesting to conside the Reidemeiste tosion of the complexes CN and C ~X. Fo cetain ings R, including Z H Š and L, ifc is a nite fee chain complex ove R with a chosen basis b, then we can de ne the Reidemeiste tosion t C b A Q R, see Appendix A. The Novikov complex CN and equivaiant cell-chain complex C ~X have natual bases consisting of lifts of the citical points o simplices of the tiangulation fom X to ~X. We denote the esulting tosion invaiants by T m A Q L =GH; T ~X A Q Z H Š =GH:

6 214 M. Hutchings We have to mod out by GH because of the ambiguity in choosing lifts and odeing the bases. It tuns out that one can esolve the fome ambiguity by choosing an Eule stuctue, see Appendix B. The space E X of Eule stuctues is a natual a½ne space ove H 1 X de ned by Tuaev. We thus obtain e ned tosion invaiants, which ae H 1 X -equivaiant maps T m : E X!Q L =G1; T ~X : E X!Q Z H Š =G1: Results in [Tu2] show that the e ned topological tosion T ~X depends only on the coveing ~X! X, and not on the choice of tiangulation. Fo example, when X is the 3-manifold obtained fom 0-sugey on a knot K, the invaiant T ~X is elated to the Alexande polynomial of K, see e.g. [Tu1, HL1]. By contast, the Mose-theoetic tosion T m depends on the admissible pai a; g, if aš is xed and nonzeo. (See Example 1.7.) To get a topological invaiant, we must multiply by the zeta function. De nition 1.5. [HL2] We de ne I :ˆ T m z A Q L =GH, and I :ˆ T m z : E X!Q L =G1: Remak 1.6. In the est of this pape, in any equation involving the Reidemeiste tosions T ~X and T m o the invaiant I, it is to be undestood that thee is an implicit `G' sign. One can use a homology oientation of X to emove the sign ambiguity in the topological tosion T ~X (see [Tu1]), and pesumably in the Mosetheoetic tosion T m as well, but we will not go into that hee. 1.6 The main esults and basic examples Ou main esults ae Theoems A and B below. These wee poved in [HL2] (genealizing [HL1]) when the cohomology class of a is a eal multiple of an integal class. A elated esult was poved by Pajitnov [Pa3] at about the same time as [HL2]; the connection of this esult with ou wok is discussed in 5. Theoem A. Fo a; g admissible, the Mose theoy invaiant I depends only on the cohomology class aš A H 1 X; R and the choice of coveing ~X. (If we change the cohomology class aš, aside fom multiplying it by a positive eal numbe, then the Novikov ing L changes, and also the choice of coveing ~X may no longe be valid, so we geneally cannot diectly compae the invaiants I. See 4 fo an exception.) We can identify the invaiant I as follows. The natual inclusion Z H Š!L induces an inclusion of quotient ings { : Q Z H Š! Q L. (To see this, one must check that the inclusion Z H Š!L sends nonzeodivisos to nonzeodivisos. This follows fom a ``leading coe½cients'' agument o fom Lemma A.4.) We then have:

7 Reidemeiste tosion 215 Theoem B. If a; g is admissible, then the Mose theoy invaiant I agees with the topological Reidemeiste tosion: I ˆ { T ~X A Q L =GH: We will also sketch a poof that the e ned invaiants agee, i.e. that 1:5 I ˆ { T ~X : E X!Q L =G1: Of couse (1.5) implies Theoem A, since T ~X is a topological invaiant. Howeve ou stategy will be to pove Theoem A st, and then deduce Theoem B and (1.5). Example 1.7. Suppose X ˆ S 1 and aš 0 0. We take ~X ˆ R, so that L F Z t. It is then not had to check the following: If a has no citical points, then T m ˆ 1 and z ˆ 1 t 1.Ifa has citical points, then T m ˆ 1 t 1 and z ˆ 1. In any case, T ~X ˆ 1 t 1. Example 1.8. Suppose a ˆ df with f : X! R a Mose function. Then thee ae no closed obits, so z ˆ 1. In this case it is classical that T m ˆ T ~X, cf. [Mi1]. (Note that { is the identity map in this case.) Hee is a sketch of a poof that in fact T m ˆ T ~X (cf. [HL2]). Choose a tiangulation T and let v T be the associated vecto eld as in Appendix B. One can appaently nd a Mose function f T and a metic g T such that the gadient g 1 T df T is a petubation of v T, so that we have a natual isomophism of chain complexes CN F C ~X, especting the bases detemined by an Eule stuctue. Then (1.5) holds fo df T ; g T, and by Theoem A it holds fo all exact 1-foms. Example 1.9. Suppose a ˆ df whee f : X n! S 1 is a be bundle with connected bes. In paticula, thee ae no citical points. Let ~X be the in nite cyclic cove as in Example 1.3, so that L F Z t. Let S be a be, and let f : S! S be the etun map obtained by following the ow V fom S though X and back to S. In this case the zeta function counts xed points of f and its iteates with thei Lefschetz signs: 1:6 z ˆ exp Py kˆ1 afix f k tk k : Thee is a canonical Eule stuctue x 0 ˆ i 1 V 0 (see Appendix B), and T m x 0 ˆ1. One can also show (cf. [F2, HL2]) that T ~X x 0 ˆ Qn 1 iˆ0 i 1 1 det 1 th i f whee H i f is the induced map on H i S; Q. So (1.5) gives hee

8 216 M. Hutchings exp Py afix f k tk k ˆ Qn 1 det 1 th i f 1 i 1 : kˆ1 iˆ0 This is equivalent to the Lefschetz theoem fo f k, as one can see by taking the logaithmic deivative of both sides. If we choose a lage coveing ~X, we obtain an equivaiant vesion of the Lefschetz theoem [F2, H]. Remak The elation between tosion and the zeta function in this example goes back to Milno [Mi2], and was extended to count closed obits of cetain nonsingula ows by Fied [F1]. The vesion of the zeta function in equation (1.6) goes back to Weil [We]. Example When X is an oiented 3-manifold with b 1 X > 0, we conjectued in [HL1], based on Taubes' wok [Ta1, Ta2, Ta3], that the invaiant I equals a cetain epaametization of the Seibeg-Witten invaiant of X. In [HL2], we combined this conjectue with Theoem B to deive a fomula fo the Seibeg-Witten invaiant of X in tems of topological tosion, which had been conjectued by Tuaev [Tu3]. This esult was late independently poved by Tuaev [Tu4], e ning a esult of Meng and Taubes [MT], and indiectly veifying the conjectue in [HL1]. Fo additional details and efeences see [HL1, HL2]. Moe ecently, using ideas fom TQFT, a pape by Donaldson [D] has appeaed giving an altenate appoach to the Meng-Taubes fomula, and T. Mak [Ma] has given a moe diect poof of most of the conjectue in [HL1]. 1.7 Ideas of the poofs The stategy fo the poof of Theoem A is to analyze the e ect on T m and z as we defom the pai a; g, xing the cohomology class aš. As long as the pai a; g emains admissible, the Novikov complex and zeta function do not change. But in a geneic 1-paamete family, the following types of bifucations (failues of admissibility) may occu: (1) A degeneate ow line fom ~p A C ~ i to ~q A C ~ i 1, (2) A degeneate closed obit, (3) A ow line fom ~p A ~C i to ~q A ~C i, whee p ~p 0 p ~q, (4) A ow line fom ~p to h~p, whee h A H, (5) Bith o death of two citical points at a degeneate citical point. The st two bifucations change neithe the Novikov complex no the zeta function. This follows fom compactness aguments fo the moduli spaces of closed obits and ow lines. Actually bifucation (2) includes not only simple cancellation of closed obits of opposite sign, but also peiod-doubling bifucations. Thus it is impotant that the closed obits ae ``counted coectly'' by the zeta function (1.1), see Remak 3.3.

9 Reidemeiste tosion 217 The thid bifucation does not a ect the zeta function, but it does change the Novikov complex, e ectively pefoming a change of basis in which ~p is eplaced by ~p G ~q. This does not change T m because the change of basis matix has deteminant one, cf. Poposition A.5. In the last two bifucations, z and T m both change, due to the inteaction of closed obits with citical points. In bifucation (4), a closed obit homologous to h is ceated o destoyed, intuitively because the ow line fom ~p to h~p is a ``boken closed obit'', which should constitute a bounday point in the one-dimensional moduli space of closed obits as time vaies. As a esult, the zeta function is multiplied by a powe seies 1 G h O h 2. At the same time, ou undestanding of bifucation (3) suggests that a change of basis occus in the Novikov complex in which ~p is multiplied by a powe seies 1 G h O h 2. Consequently the tosion T m is multiplied by this powe seies o its invese. We nd in this way that I is unchanged ``to st ode''. The highe ode tems ae moe di½cult to undestand, essentially because the defomation upstais in ~X is not geneic, due to its H-equivaiance, so that thee ae multiply boken closed obits and ow lines at the time of bifucation. It appeas that z and T m ae simply multiplied by seies of the fom 1 G h G1. But instead of tying to pove this, we conside a non-equivaiant petubation of the defomation in a nite cyclic cove ^X! X. The idea is that invaiance to st ode in ^X implies invaiance to highe ode in X, which we pove afte woking out the behavio of the invaiant I with espect to nite cyclic coves. In this way we show that I is unchanged by a bifucation of type (4). Bifucation (5) also has the subtlety of multiply boken ow lines and closed obits, aising fom concatenations of ow lines fom the degeneate citical point to itself. Hee we use diect nite dimensional analysis to show that evey multiply boken closed obit o ow line leads to an unboken closed obit o ow line on the side of the bifucation time whee the two citical points die, but not on the othe side. Some miaculous algeba then yields that I is invaiant. A small complication in the agument outlined above is that the times at which bifucations occu might not be isolated. But bifucations involving ``shot'' closed obits o ow lines ae isolated, and the long bifucations change only ``highe ode'' tems in I. Taking a limit in which we conside successively longe bifucations, we conclude that I is invaiant. With Theoem A established, Theoem B can be deduced athe easily. We aleady saw in Example 1.8 that Theoem B holds when a is exact. Fo geneal a, we use a tick of F. Latou which allows us to ``appoximate'' a by an exact 1-fom (!). Namely, we let f : X! R be a Mose function and we eplace a with the cohomologous fom b ˆ a Cdf fo C A R lage. The Novikov complex and zeta function of b ae the same as those of the escaled fom

10 218 M. Hutchings C 1 b ˆ df C 1 a: Fo C lage, thee ae no closed obits, and the Novikov complex of b coincides with that of df (unde an inclusion of Novikov ings). So by Example 1.8, Theoem B holds fo b, and by invaiance (Theoem A) it holds fo a as well. 2 Poof of invaiance I: setup In this section we make some geneal pepaations fo the poof of Theoem A by bifucation analysis. In 3 we will undetake the analysis of speci c bifucations and complete the poof of Theoem A. 2.1 Semi-isolated bifucations Conside a 1-paamete family f a t ; g t g of 1-foms and metics, paametized by t A 0; 1Š. A geneic family may have a countably in nite set of bifucations. In this section we set up a famewok in which we only need to conside one bifucation at a time. Moe pecisely, Lemma 2.7 makes sense of the change in I caused by a single bifucation, and Lemma 2.9 shows that if all these individual changes ae zeo, then I is invaiant. Note that we always assume that the cohomology class a t Š is independent of t. De nition 2.1. A ow line between two citical points is degeneate if it coesponds to a nontansvese intesection of ascending and descending manifolds. A (k times) boken ow line fom ~p to ~q is a concatenation of ow lines fom ~p to ~ 1 to ~ 2 to... to ~ k to ~q, whee ~ 1 ;...; ~ k ae citical points and k V 1. A boken closed obit in the homology class h is a (possibly boken) ow line fom ~p to h~p fo some citical point ~p. Let M t ~p; ~q denote the space of (unboken) ow lines fom ~p to ~q at time t. Let O t h denote the space of (unboken) closed obits homologous to h at time t. If the zeoes of a t ae nondegeneate fo all t A t 1 ; t 2 Š, then thee is a canonical identi- cation of citical points ~ C t ˆ ~C t 0 fo any t; t 0 A t 1 ; t 2 Š, which we make implicitly below. The following lemma implies that ou invaiant does not change if thee ae no bifucations, as a esult of suitable compactness. Lemma 2.2. Let t 1 < t 2. Suppose a t ; g t is admissible at t 1 and t 2. (a) Suppose thee ae no degeneate citical points fo t A t 1 ; t 2 Š. Let ~p A ~C i and ~q A ~C i 1. Suppose thee ae no degeneate o boken ow lines fom ~p to~q fo t A t 1 ; t 2 Š. Then M t1 ~p; ~q ˆM t2 ~p; ~q :

11 Reidemeiste tosion 219 (b) Let h A H. Suppose thee ae no degeneate o boken closed obits homologous to h fo t A t 1 ; t 2 Š. Then O t1 h ˆO t2 h : Moeove, the above bijections ae oientation peseving. Poof. (a) We st claim that S t A t 1 ; t 2 Š M t ~p; ~q is compact. To see this, let S ~p be a small sphee in the descending manifold aound ~p, and let S ~q be a small sphee in the ascending manifold aound ~q. A ow line coesponds to a tiple x; y; s A S ~p S ~q R such that downwad ow fom x fo time s hits y. Compactness will follow fom an uppe bound on s. Ifs is unbounded, then one can show as in 1.3 that thee is a boken ow line fom ~p to ~q at some time t A t 1 ; t 2 Š, contadicting ou hypothesis. Now (possibly afte a petubation as in 2.2), S t A t 1 ; t 2 Š M t ~p; ~q is a compact onemanifold with bounday M t2 ~p; ~q M t1 ~p; ~q. Moeove thee ae no cancellations, since thee ae no degeneate ow lines fom ~p to ~q fo t A t 1 ; t 2 Š. Fo pat (b), we get a simila compactness fo S t A t 1 ; t 2 Š O t h, as in 1.3, since thee ae no boken closed obits. Since the obits emain nondegeneate, none ae ceated o destoyed, and the Lefschetz signs cannot change. De nition 2.3. A bifucation of the family f a t ; g t g is a time t 0 A R such that the pai a t0 ; g t0 fails to be admissible. Fo nonexact closed 1-foms, a geneic one-paamete family may contain in nitely many bifucations (fo basically the same eason that a geneic nonexact closed 1- fom may have in nitely many closed obits and ow lines between citical points of index di eence one). Howeve a geneic one-paamete family will only contain nitely many bifucations with a given uppe bound on the ``length''. De nition 2.4. The length of a bifucation t 0 is the smallest of the following numbes: (a) 0, if a t0 has a degeneate zeo. (b) a t0 Š h, whee h is the homology class of a degeneate o boken closed obit. (c) g a t 0, whee g is a degeneate (downwad) ow line. Lemma 2.5. Fo any family f a t ; g t g and any R A R, the set of bifucations t A 0; 1Š of length U R is closed. Poof. Let ft n g be a sequence of bifucations conveging to t 0. Afte passing to a subsequence, we may assume that each of the bifucations t n includes the same type of degeneate object (a citical point, a closed obit, o a ow line between two given citical points in X ) with length U R. By a compactness agument as in 1.3, these objects accumulate to an object of the same type (possibly boken) at time t 0 with length U R. Since tansvesality is an open condition, this object at time t 0 is also degeneate (o boken), so t 0 is a bifucation of length U R.

12 220 M. Hutchings We now digess to intoduce the notion of limits in Novikov ings. Given x ˆ P g a g g A Nov G; N and R A R, we wite ``x ˆ O R '' if a g ˆ 0 wheneve N g < R. Given a sequence fx n g in Nov G; N and x A Nov G; N, we wite ``lim n!y x n ˆ x'' if fo evey R A R thee exists n 0 such that x x n ˆ O R fo all n > n 0. We can extend these de nitions to the quotient ing Q L as follows. If G is a nitely geneated abelian goup, then by Lemma A.4 we have a decomposition Q Nov G; N ˆ L F j into a sum of elds. By (A.3), each eld F j can be identi ed with the tenso poduct of Nov G=Ke N ; N with a cetain eld. The notion of ``O R '' is then well de ned fo elements of F j. We say that an element of Q Nov G; N is ``O R '' if its pojection to each sub eld F j is O R, and we de ne limits accodingly. De nition 2.6. A time t 0 is good if: (a) t 0 is not a limit of bifucations of bounded length. (b) Fo each e > 0, the intevals t 0 e; t 0 and t 0 ; t 0 e both contain some times t which ae not bifucations. Lemma 2.7. If t 0 is good, then the limits as t % t 0 and t & t 0 of z and CN ; q ae well de ned. If moeove t 0 is not a bifucation, then the left and ight limits of z (esp. CN ) ae both equal to z t 0 (esp. CN t 0 ). Poof. Conside the limit as t % t 0. Thee exists e > 0 such that all citical points of a t ae nondegeneate fo t A t e; t 0, so that ~C t ˆ ~C t 0 fo t; t 0 A t 0 e; t 0. Fo convegence of q, we must show that fo ~p A ~ C i and ~q A ~ C i 1, thee exists x A L such that lim t%t 0 P h ~p; h~qi h ˆ x; h A H whee t anges ove any sequence of non-bifucation values conveging to t 0 fom below. We use Lemma 2.2(a). Fo any path g fom ~p to h~q, we have g a ˆ C aš h whee C is a constant which is independent of h and vaies continuously with t. Thus if g is a downwad gadient ow line and aš h is bounded fom above then g a is also bounded fom above. So if we ae su½ciently close to t 0 then thee ae no degeneate o boken ow lines fom ~p to h~q by de nition of semi-isolated, so h ~p; h~qi cannot change by Lemma 2.2(a). Similay, Lemma 2.2(b) implies that the zeta functions convege. The last sentence of the lemma follows fom Lemma 2.2.

13 Reidemeiste tosion 221 Let z, z, CN ; q, CN ; q denote these limits. An Eule stuctue gives a basis fo the limiting complexes CN and CN ; let T m, T m denote thei Reidemeiste tosion, and let I G :ˆ Tm G zg. Lemma 2.8. If t 0 is good, then fo any Eule stuctue x, lim I t x ˆI t 0 x ; lim I t x ˆI t 0 x ; t%t 0 t&t0 whee t anges ove non-bifucations. Poof. Conside the limit as t % t 0. By de nition we have lim t%t0 z t ˆz t 0.So we have to pove that lim t%t0 T m t x ˆT m t 0 x. Fo e su½ciently small we can identify the citical points fo di eent t A t 0 e; t 0. Fix a basis fo CN consisting of a lift of each citical point to ~X, in the equivalence class detemined by x. Fo a non-bifucation t, ecall that T m is the sum of the tosions of CN n F j. The tosion of CN n F j is zeo if CN n F j is not acyclic; this citeion is independent of t, by the Novikov isomophism (1.4). Moeove, even if t 0 is a bifucation, the limiting complex CN n F j is acyclic if and only if CN n F j is acyclic fo all nonbifucations t, because the Novikov isomophism (1.4), as constucted in [HL1], can be extended by a limiting agument to CN. When CN n F j is acyclic, we compute its tosion using Poposition A.2. Choose subbases D i and E i as in Poposition A.2 fo CN n F j. We can use these same subbases in the inteval t 0 d; t 0 fo some d, because if the deteminants in Poposition A.2 ae nonzeo in the limiting complex, then they ae nonzeo nea t 0. The eason is that each deteminant fo the limiting complex has a nonzeo ``leading tem'' involving ow lines of length < R fo some R, which will be unchanged nea t 0 by Lemma 2.2(a). Fo a; b A F j we have 1 a 1 ˆ O R when the leading ode of a b exceeds the b leading ode of a and b by at least R. This means that a high ode change in the denominato of T m x, as computed above, will change T m x by high ode tems. We ae now done by condition 2.6(a) and Lemma 2.2(a). Lemma 2.9. Let f a t ; g t g be a family paametized by t A 0; 1Š, with a t in a xed cohomology class and a 0 ; g 0 and a 1 ; g 1 admissible. Suppose that evey bifucation t 0 A 0; 1 is good and satis es I t 0 ˆI t 0. Then I 0 ˆI 1. Poof. We st obseve that evey t 0 A 0; 1Š is good. If t 0 fails to satisfy condition 2.6(a), then t 0 is a bifucation by Lemma 2.5, contadicting ou hypothesis that all bifucations ae good. Since we also assumed that evey bifucation satis es condition 2.6(b), it follows that the non-bifucations ae dense in 0; 1Š, so evey t 0 A 0; 1Š satis es condition 2.6(b). Next, by the assumptions and Lemma 2.7, we have I t 0 ˆI t 0 fo each t 0 A 0; 1Š. It follows fom Lemma 2.8 that if we x an Eule stuctue x and R > 0, then fo all t 0 A 0; 1Š, thee exists e > 0 such that

14 222 M. Hutchings I t x ˆI t 0 x O R fo all non-bifucations t; t 0 A t 0 e; t 0 e. Since 0; 1Š is compact and the nonbifucations ae dense, it follows that I 0 x I 1 x ˆO R. Taking R! y, while keeping x xed, completes the poof. To educe complications, we will conside good bifucations which satisfy an additional condition. De nition A bifucation t 0 is semi-isolated if t 0 is good, and: (*) The pai a t0 ; g t0 violates only one of the admissibility conditions in De nition 1.1, and at only one degeneate object. 2.2 Geneic one-paamete families The following lemma implies that in a geneic one-paamete family, only the ve types of bifucations listed in 1.7 may occu, and ae semi-isolated. Lemma Let f a t ; g t ; t A 0; 1Šg be a 1-paamete family with a t in a xed cohomology class and a 0 ; g 0 and a 1 ; g 1 admissible. Then afte a petubation xing the endpoints, we may aange that: (a) Nea a degeneate citical point at time t 0, thee ae local coodinates x 1 ;...; x n in which 2:1 g 1 t a t ˆ x 2 1 G t t 0 ; x 2 ;...; x i ; x i 1 ;...; x n : (b) Suppose that fo t A t 1 ; t 2 Š, we have citical points ~p t ; ~q t A ~C a t which depend continuously on t. Then S t D ~p t and S t A ~q t intesect tansvesely in ~X t 1 ; t 2 Š, and the pojection of the intesection to t 1 ; t 2 Š is a Mose function. (c) All bifucations ae semi-isolated. If thee ae no degeneate citical points in the oiginal family a t ; g t, then we may choose this petubation to be C k small fo any k. Poof. We will wok with C k families so that we can use the Sad-Smale theoem. Afte aanging (a), we will show that in the space of C k families, thee is a countable intesection of open dense sets, whose elements ae families with the desied popeties (b) and (c). As in [MS, Ta2], we then obtain a dense set in C y. We begin by making the gaph of S t a t tansvese to the 0-section of T X 0; 1Š. Then S t a 1 t 0 is a smooth 1-dimensional submanifold of X 0; 1Š. We can futhe aange that t is a Mose function on S t a 1 t 0 such that all citical points have distinct values. A citical point of t on S t a 1 t 0 is a pai x; t whee a t has a degeneate zeo at x. By a lemma of Cef [Ce] we can choose (possibly time dependent) local coodinates x 1 ;...; x n nea such a point so that

15 Reidemeiste tosion 223 a t ˆ x3 1 3 G t t 0 x 1 x2 2 x2 i xi 1 2 x2 n : 2 We now x the metic g t on X to be Euclidean nea the oigin in these coodinates. This gives (a). By a standad tansvesality agument, we can obtain (b) in a countable intesection of open dense sets. Fixing the metic nea the degeneate citical points does not intefee with the tansvesality agument because no ow line o closed obit is completely suppoted nea a degeneate citical point. To obtain (c), we st aange fo the space of ieducible closed obits in X 0; 1Š to be cut out tansvesely (egaded as the zeo set of a section of a vecto bundle as in Remak 3.3), fo the pojection fom this space to 0; 1Š to be a Mose function, and fo the degeneate coves with a given multiplicity of a given connected family of obits to be isolated. We then use a compactness agument as in Lemma 2.5 to show that (i) fo each R, only nitely many degeneate objects of length < R occu. We can aange that these degeneate objects occu at distinct times, by intesecting with an open dense set of defomations. So in a countable intesection of open dense sets, we can aange that (ii) all degeneate objects occu at distinct times. Now (i) and (ii) imply (c). 3 Poof of invaiance II: bifucation analysis In a geneic one-paamete defomation given by Lemma 2.11, only the ve types of bifucations listed in 1.7 may occu, and all bifucations ae semi-isolated. In this section we will show that I ˆ I fo each such bifucation. By Lemma 2.9, this will complete the poof of Theoem A. 3.1 Cancellation of ow lines Lemma 3.1. Suppose t 0 is a semi-isolated bifucation at which thee is a degeneate ow line fom ~p A ~ C i to ~q A ~ C i 1. Then z t 0 ˆz t 0 and CN ; q ˆ CN ; q. Poof. By the de nition of semi-isolated, we may choose e > 0 such that fo all t with 0 < jt t 0 j U e, thee ae no degeneate o boken ow lines fom ~p to ~q. Asinthe poof of Lemma 2.2(a), the moduli space of ow lines fom ~p to ~q fo jt t 0 j U e is compact, so hq ~p; ~qi ˆ hq ~p; ~qi, since the signed numbe of bounday points of a compact 1-manifold is zeo. Fo evey R > 0, fo evey othe pai of citical points ~; ~s with index di eence 1 and ~ ~s a < R, the coe½cient hq~; ~si likewise does not change fo t su½ciently close to t 0. Fo evey R > 0, the coe½cients in the zeta function of h with aš h < R do not change fo t su½cently close to t 0, by Lemma 2.2(b). 3.2 Cancellation of closed obits Lemma 3.2. Suppose t 0 is a semi-isolated bifucation at which thee exists a degeneate closed obit. Then CN ; q ˆ CN ; q and z ˆ z.

16 224 M. Hutchings Poof. The Novikov complex is unchanged as in the poof of Lemma 3.1. To show that the zeta function does not change, the idea is that locally the zeta function looks like (1.6), and this is invaiant because the signed numbe of xed points of a map is invaiant, assuming suitable compactness. Moe pecisely, at time t 0 thee is an isolated ieducible closed obit g, with gš ˆh, such that g o some multiple cove of it is degeneate. Choose x A g S 1 H X, and let D d H X be a disc of adius d tansvese to g and centeed at x. Let f d; t : D d! D d be the (patially de ned) st etun map fo the ow g 1 t a t. We estict the domain of f d; t to a maximal connected neighbohood of x on which it is continuous. De ne z d; t :ˆ exp Py kˆ1 h k k afix fk d; t fo non-bifucations t. We claim that 3:1 z lim t&t0 z d; t ˆ lim : z d!0 lim t%t0 z d; t To pove this, given R > 0, we must nd d > 0 such that z z ˆ lim t&t 0 z d; t O R. By lim t%t0 z d; t the de nition of semi-isolated, thee exists e > 0 so that fo jt t 0 j < e, all closed obits g 0 with aš g 0 Š < R ae nondegeneate, except fo coves of g at time t ˆ t 0. By compactness as in Lemma 2.2(b), we can choose d su½ciently small that no such closed obit (othe than coves of g) intesects D 2d at time t 0. Then fo jt t 0 j su½ciently small, the contibution to log z fom closed obits g 0 avoiding D d with aš g 0 Š < R does not change, and when moeove t is not a bifucation, the contibution to log z fom all othe closed obits g 0 with aš g 0 Š < R is counted by the ode < R tems of log z d; t, as in (1.6). This poves (3.1). Given any positive intege k, as above we can choose d such that at time t 0,no closed obit g 0 with aš g 0 Š U k aš h (othe than coves of g) intesects D 2d.In paticula, fo k 0 U k, the bounday of the gaph of f k 0 d; t 0 does not intesect the diagonal in D d D d. (Hee we ae compactifying the gaph as in the poof of Lemma 3.8(b), see also [HL1].) It follows that afix f k 0 d; t is independent of t fo nonbifucations t close to t 0. This implies that lim t&t0 z d; t lim ˆ 1: d!0 lim t%t0 z d; t Togethe with (3.1), this poves the lemma. Remak 3.3. Hee ae two altenate appoaches to poving this lemma, which add some insight and might genealize to Floe theoy. Fist, one might show that geneically thee is eithe a simple cancellation of two obits, o a peiod doubling bifucation coesponding to 1 h ˆ 1 h 2 1 h 1

17 Reidemeiste tosion 225 in the poduct fomula (1.2). Related analysis appeas in [Ta2] fo the moe complicated poblem of counting pseudoholomophic toi in symplectic 4-manifolds. Second, one might make the following fomal agument igoous. Fo nonzeo h A H, let L h denote the space of loops g : S 1! X homologous to h, modulo otations of S 1. Thee is a vecto bundle E ove L h R given by E g; l ˆ G g TX : De ne a section s of E by s g; l t ˆg 0 t lv g t : By de nition g is a closed obit if and only if s g; l ˆ0 fo some (unique) l > 0. So the coe½cient of h in log z, namely 3:2 P g A O; gšˆh 1 m g p g A Q; is fomally the degee of the section s. We divide by p g because L h is an obifold with Z=p symmety aound loops with peiod p. As long as thee is no inteaction between closed obits and citical points, so that l stays bounded and the zeo set of s emains compact, the coe½cients (3.2) of log z, and hence z, should not change. 3.3 The slide bifucation A slide bifucation is a semi-isolated bifucation t 0 at which thee is a downwad ow line fom ~p A ~C i to ~q A ~C i. (Fo eal-valued Mose functions, this bifucation acts on the coesponding handle decomposition of X by sliding one handle ove anothe.) By Lemma 2.11, we can (and will) assume that the ow line fom ~p to ~q is a tansvese intesection of S t D ~p t and S t A ~q t. Lemma 3.4. Fo a slide bifucation such that p ~p 0 p ~q in X, we have (a) z ˆ z and q ˆ A 1 q A, whee A : CN! CN sends ~p 7! ~p G ~q and xes all othe citical points ~s with p ~s 0 p ~p. (b) In paticula I ˆ I. Poof. Since the bifucation is semi-isolated, if ~ is a citical point with ind ~ 0 i 1 and p ~ 0 p ~p, then 3:3 q ~ ˆq ~ : Let us denote this common value by q 0 ~. We now claim that 3:4 q ~p ˆq ~p G q 0 ~q :

18 226 M. Hutchings To see this, let ~s be a citical point of index i 1. We need to show that 3:5 hq ~p; ~si ˆ hq ~p; ~sighq~q; ~si; with the G sign independent of ~s. (This fomula makes sense because hq~q; ~si is independent of t nea the bifucation time t 0.) One way to pove equation (3.5) (fom [Lau]) is as follows. Choose f : ~X! R with df ˆ p a, and let S be a level set of f just below ~q. Conside the ``descending slices'' D ~p ˆD ~p X S and D ~q ˆD ~q X S, which depend on t nea t 0. These can be given the stuctue of chains [HL1] o cuents [Lau]. Let D ~p and D ~p denote the limits of D ~p as t appoaches t 0 fom above and below, and let D 0 ~q ˆ D ~q t 0. We then have, as chains, 3:6 D ~p ˆD ~p G D 0 ~q ; see [Lau]. Taking the intesection numbe with A ~s, we obtain (3.5). Anothe appoach to poving (3.5), which genealizes to in nite dimensions, is to use a gluing agument as in [Fl] to show that fo e small, we have a one-dimensional cobodism q S t A t 0 e; t 0 eš! M t ~p; ~s ˆ M t0 e ~p; ~s M t0 e ~p; ~s H M t0 ~p; ~q M t0 ~q; ~s : (The oientations ae moe subtle in this appoach. Related gluing aguments can also be used to pove (3.6), cf. [HL1].) Similaly to (3.4), fo each citical point ~s of index i 1, we have 3:7 q ~s ˆq ~s Hhq~s; ~pi~q: Equations (3.3), (3.4) and (3.7) imply (a). Pat (b) follows by Poposition A.5, since det A i ˆ Tosion and zeta function of a nite cyclic cove We now digess to wok out the behavio of the invaiant I with espect to nite cyclic coves. The answe is given in tems of the Nom map fom Galois theoy. This esult will be needed when we use nonequivaiant petubations in the next section, and may also be of independent inteest. Suppose we have a shot exact sequence of abelian goups 0! K! { H! m Z=k! 0: Let : ^X! X be the k-fold cyclic coveing whose monodomy is the composition p 1 X!H! m Z=k. The coveing ~X! X factos though, and the coveing

19 Reidemeiste tosion 227 ~X! ^X has automophism goup K. We now want to elate the invaiants of X and ^X, choosing the coveing ~X fo both in Choice 1.2. We need the following algebaic notation. Let ^L :ˆ Nov K; { aš. The map { induces a pushfowad of Novikov ings { : ^L! L sending P k A K a k k 7! P Pk A K a k { k. Since { has nite kenel, thee is also a pullback { : L! ^L sending h A H a h h 7! P k A K a { k k. The pushfowad { makes L into a fee module of ank k ove ^L. Ify A L, then multiplication by y is an endomophism of this module, whose deteminant and tace we denote by Nom y and T y espectively. It will sometimes be convenient to assume that: 3:8 m annihilates the tosion subgoup of H: In geneal, the map { sends nonzeodivisos to nonzeodivisos and hence induces a map on quotient ings Q ^L!Q L. Recall fom Lemma A.4(a) that we have decompositions of Q ^L and Q L into sums of elds. Assumption (3.8) implies that { espects these decompositions. We then see fom (A.3) that Q L is a fee module of ank k ove Q ^L, so Nom extends to a multiplicative map Q L!Q ^L. Lemma 3.5. (a) If y A L then T y ˆk { y. (b) If x A L, then log Nom 1 x ˆT log 1 x. (c) Assuming (3.8), ifyaq L and y 0 0, then Nom y 0 0. Poof. Let y be a pimitive k th oot of 1. Fo 0 U i < k, de ne a ing homomophism s i : L n Z yš!l n Z yš by s i h n 1 ˆh n y im h fo h A H. By [Lan, 6.5], we have 3:9 T y ˆ Pk 1 s i y ; iˆ0 Nom y ˆ Qk 1 s i y : iˆ0 The st of these identities implies that fo h A H, T h ˆ kh if m h ˆ0 0 if m h 0 0 (which can also be seen moe diectly). This poves (a). To pove (b), we use (3.9) to compute that log Nom 1 x ˆ Pk 1 log s i 1 x ˆ Pk 1 s i log 1 x ˆT log 1 x : iˆ0 Hee the middle equality holds because log is de ned using a powe seies (see Remak 1.4) and s i is a ing homomophism. To pove (c), obseve that assumption (3.8) implies that s i espects the eld decomposition of Q L. Assetion (c) now follows fom (3.9) and the injectivity of s i. iˆ0

20 228 M. Hutchings If V is a vecto eld on X with nondegeneate zeoes, then the invese image map H 1 X; V!H 1 ^X; V induces a natual pullback of Eule stuctues : E X! E ^X. It is clea that if a; g is admissible on X, then a; g is admissible on ^X. With espect to these pullbacks, the invaiant I behaves as follows. Poposition 3.6. (a) z ^X ˆNom z X. (b) Unde the assumption (3.8), the following diagam commutes: E ^X x??? T m ^X ƒƒƒ! T m X E X ƒƒƒ! Q ^L =G1 x??? Nom Q L =G1: Poof. (a) Evey closed obit ^g in ^X is a lift of a unique closed obit g in X, with gš A K. Convesely, if g A O X and gš A K, let g 1 denote the peiod one obit undelying g, and let l be the ode of m g 1 Š in the goup Z=k. Then g lifts to k=l distinct closed obits ^g, each of which has peiod p ^g ˆp g =l and Lefschetz sign 1 m ^g ˆ 1 m g. Theefoe log z ^X ˆ By Lemma 3.5, P ^g A O ^X 1 m ^g p ^g ^gš ˆ P g A O X ; gš A K k{ log z X ˆT log z X ˆlog Nom z X : k 1 m g gš ˆk{ log z X : p g Combining the above equations and applying exp poves (a). (b) A nite fee complex C ove L can be egaded as a complex ^C ove ^L with k times as many geneatos. Moeove, a basis fl 1 ;...; l k g fo L ove ^L detemines a map f : B C!B ^C, and if w C ˆ0 then the map f is independent of the choice of fl 1 ;...; l k g. Now we obseve that if x A E X, then the Novikov complex CN ^X, with the basis detemined by x, is obtained fom CN X and x by this constuction. So we need to show that t ^C f b ˆ Nom t C b. The assumption (3.8) implies that { and Nom ae compatible with the decompositions of Q ^L and Q L into sums of elds. So we can estict attention to a complex C n F whee F H Q L is a eld; let ^F denote the coesponding eld in Q ^L. IfC n F is not acyclic, then ^C n ^F is not acyclic eithe, so both tosions ae zeo. If C n F is acyclic, we can decompose it into a diect sum of 2-tem acyclic complexes. Ou claim then educes to the fact that if q is a squae matix ove F and ^q is the coesponding matix ove ^F, then det ^q ˆNom det q. This follows fom the de nition of Nom, afte putting q into Jodan canonical fom ove an algebaic closue of F.

21 Reidemeiste tosion Sliding a citical point ove itself We now analyze bifucation (4), in which a citical point slides ove itself, following the stategy descibed in 1.7. If p A C and x A L, let A p x : CN! CN denote the L-module endomophism which sends ~p 7! x~p fo evey lift ~p of p and xes all othe citical points ~s with p ~s 0 p ~p. Lemma 3.7. Suppose ~p A ~C i slides ove h~p fo some h A H, and let p ˆ p ~p. Then (a) Thee is a powe seies x ˆ 1 P y nˆ1 a nh n, with a n A Z, such that 3:10 q ˆ A p x 1 q A p x : (b) In paticula T m ˆ x 1 i Tm. (c) The coe½cient a 1 ˆ G1. Poof. (a) Let d denote the divisibility of h in H. (This is de ned because h is not a tosion class.) Let k be a positive intege elatively pime to d, and let m : H! Z=k be a homomophism sending h 7! 1. Let : ^X! X be the k-fold cyclic cove with monodomy m. Then the citical points ~p; h~p;...; h k 1 ~p poject to distinct points in ^X. Let R ˆ aš kh. By semi-isolatedness, we can nd e > 0 such that no bifucation of length < R occus between time t 0 e and t 0 e, othe than the slide of ~p ove h~p. Choose a smalle e if necessay so that the pais a t0 Ge; g t0 Ge ae admissible. Petub the pulled back family f a t ; g t jt A t 0 e; t 0 ešg, xing the endpoints, to satisfy the geneicity conditions of Lemma By a compactness agument (as in the poof of Lemma 2.5), we can choose the petubation small enough that no bifucations of length < R occu othe than slides of h i ~p ove h j ~p. Then iteating Lemma 3.4(a) and using Lemma 2.2(a), we nd a powe seies x k ˆ 1 P k 1 nˆ1 a n; kh n such that 3:11 q ˆ A x k 1 q A x k O R : (Hee ``O R '' indicates a tem involving ow lines g with g a V R.) Without loss of geneality, q ~p 0 0ohq ~s; ~pi00 fo some s (since othewise equation (3.10) is vacuously tue fo any x). Then equation (3.11) implies that fo n xed, a n; k is constant fo lage k. If we de ne a n ˆ lim k!y a n; k, then equation (3.10) follows. Assetion (b) follows fom (a) and Poposition A.5. Now ecall that the slide of ~p ove h~p comes fom a single tansvese cossing of ascending and descending manifolds. Unde a su½ciently small petubation of the defomation in ^X, this cossing will pesist, and no othe such cossing will appea, by a compactness agument. So fo a su½ciently small petubation, a 1; k ˆ G1, and hence a 1 ˆ G1. This poves (c).

22 230 M. Hutchings Lemma 3.8. Suppose ~p A ~C i slides ove h~p. Then (a) Thee is a powe seies y ˆ 1 P y nˆ1 b nh n such that z ˆ y z. (b) b 1 ˆ 1 i 1 a 1. (See Lemma 3.7.) Poof. (a) By Lemma 2.2(b), a closed obit can be ceated o destoyed in the bifucation only if it is homologous to kh fo some k. So log z log z is a powe seies in h. Thus z =z is a powe seies in h. (A pioi the coe½cients b n ae ational; it's not impotant hee, but we actually know that b n A Z, due to the poduct fomula (1.2) fo the zeta function.) (b) Let Z H X be a compact tubula neighbohood of the ow line g fom p to itself at t 0. Thee is a function f : Z! R=Z such that aj Z ˆ l df fo some l A R. Let S H Z be a level set fo f away fom p. The ow V induces a patially de ned etun map f : S! S. Closed obits homologous to h nea g ae in one to one coespondence with xed points of f. A xed point of f is an intesection of the diagonal D H S S with the gaph G f, and the Lefschetz sign of the closed obit equals the sign of the intesection. The gaph G f has a natual compacti cation (see [HL1]) to a manifold with cones G whose codimension one statum is qg ˆ A p D p W Y: Hee D p and A p ae the `` st'' intesections of the descending and ascending manifolds of p with S, and Y is a component aising fom tajectoies that escape the neighbohood Z. The numbe of closed obits nea g changes wheneve D p A p cosses D. This is happening at time t 0 at a single point, tansvesely, and an oientation check shows that the sign is 1 i 1 a 1. No othe closed obits homologous to h can be ceated o destoyed, as in Lemma 2.2(b). Remak 3.9. It should also be possible to pove (b) using a Floe-theoetic gluing agument to show that in the homology class h, a single closed obit is ceated o destoyed. Lemma Suppose ~p slides ove h~p. Then I ˆ I. Poof. By Lemmas 3.7 and 3.8, we can wite 3:12 I ˆ exp Py c n h n I : nˆ2 fo some c 2 ; c 3 ;... A Q. We need to show that each coe½cient c k vanishes. Let d denote the divisibility of h in H. Let m : H! Z=dk be a homomophism which sends h 7! d and annihilates the tosion subgoup of H. Let : ^X! X be the coesponding nite cyclic cove. By Poposition 3.6 and Lemma 3.5,