Math. Z. (2012) 270:1051–1056DOI 10.1007/s00209-010-0838-7 Mathematische Zeitschrift

Floer–Novikov homology and applications to Lagrangiansubmanifolds

Mihai Damian

Received: 4 October 2010 / Accepted: 30 October 2010 / Published online: 15 January 2011© Springer-Verlag 2011

Abstract We use a non-Hamiltonian version of Lagrangian Floer homology to prove thatan exact Lagrangian submanifold in the cotangent bundle of the 3-torus T3 must be diffeo-morphic to T3. This improves a previous result of Fukaya, Seidel and Smith.

Keywords Lagrangian embeddings · Novikov homology · Floer homology ·Bieri–Neumann–Strebel invariant

Mathematics Subject Classification (2000) 57R17 · 57R58 · 53D12 · 57R80

1 Introduction

One of the most important open questions about Lagrangian submanifolds asks whether anexact Lagrangian L in the cotangent bundle of a closed manifold T ∗M can be deformed intothe zero-section by a Hamiltonian isotopy. It was raised by Arnold in [2]. The answer is notknown except for the case when M = S2—proved by Hind [15]—and seems to be out ofreach by the techniques commonly used in symplectic topology. The subsidiary problem ofthe comparison of the topologies of L and M is also quite difficult, but more intensively stud-ied. Many partial results were obtained by various authors: Audin [3], Viterbo [21], Lalondeand Sikorav [16]. An important breakthrough was made some years ago by Fukaya et al.who proved in [13]:

Theorem 1.1 Let L ⊂ T ∗M be an exact Lagrangian submanifold. Suppose that L and Mare spin and that the Maslov class of L is zero. Then the projection p : L → M induces anisomorphism in homology.

M. Damian is supported by ANR project “Floer Power” ANR-08-BLAN-0291-03.

M. Damian (B)Université de Strasbourg, IRMA, 7, rue René Descartes, 67084 Strasbourg, Francee-mail: damian@math.u-strasbg.fr

123

1052 M. Damian

More recently, Abouzaid [1] generalized the above result by showing that, under the samehypothesis, the projection p is actually a homotopy equivalence. Still, the problem of thetopological equivalence of L and M remains widely open when the Maslov class is not knownto be zero. In the present paper we give a positive answer to this question when M is thethree-torus:

Theorem 1.2 Let L ⊂ T ∗T3 be an exact Lagrangian submanifold. Then L is diffeomorphicto T3.

The result is a consequence of the following theorem, proved in our previous paper ([6],Th. 3.1):

Theorem 1.3 Consider a closed manifold M, a class u ∈ H1(M, R) and let L ⊂ T ∗Mbe a closed exact Lagrangian submanifold. Denote by p : L → M the projection on thezero-section. Suppose that there is a (closed) one-form without zeroes in the class u. Thenthe Novikov homology H∗(L; p∗u, Z/2) vanishes.

Actually, the proof given in [6] only works for u ∈ H1(M, Z) but, as we will explain in thefollowing section, it is easy to extend it for any cohomology class. The theorem is a straight-forward application of a non-Hamiltonian version of Lagrangian Floer homology which wasestablished in [6]. Namely, we build in [6] a Floer-type complex (over Z/2) spanned bythe intersection points between an exact Lagrangian L ⊂ T ∗M and its deformation by asymplectic isotopy (φt ) and we show that the homology of this complex is isomorphic to theNovikov homology of L associated to the Calabi invariant u ∈ H1(L; R) of the isotopy (φt ).On the other hand, if α is a closed one-form without zeroes, L + tα is disjoint from L if tis large enough, so this Floer–Novikov homology vanishes, which means that the Novikovhomology is also zero. For the definition and some relevant properties of Novikov homologywe refer to [5,6,19].

We finish our paper with another application of Floer-Novikov theory:

Theorem 1.4 Let L ⊂ R2n be an exact Lagrangian submanifold which equals the zero sec-tion outside a compact set. Then Hi (L , Z) = 0 for i > 0 and L is simply-connected. As aconsequence, for n �= 4 the Lagrangian L is diffeomorphic to Rn. For n = 4 the manifold Lis homeomorphic to Rn.

This is a relative (i.e. Lagrangian) version of a classical theorem in symplectic topology,known as Eliashberg–Floer–McDuff theorem [7,8], which asserts that a symplectic man-ifold which is symplectically aspherical and standard at infinity must be diffeomorphic toR2n . More precisely, if (M2n, ω) is symplectic and for some compacts K ⊂ M and K ′ ⊂ R2n

there is a symplectomorphism between M\K and R2n\K ′, then M is diffeomorphic to R2n ,provided that ω|π2(M) = 0. For 2n = 4 we have more: another celebrated theorem of Gromov[14] asserts that under this hypothesis M is actually symplectomorphic to R4.

Theorem 1.4 is an easy corollary of the result of M. Abouzaid mentionned above [1],which in addition is much more general. But in this particular case our proof is very simple.Also note that the fact that the homology of L vanishes in positive degrees was already provedby Fukaya et al. [13], again with more complicated methods.

In the remaning two sections we explain how Theorem 1.3 implies our results Th. 1.2 andTh. 1.4.

123

Floer–Novikov homology and applications to Lagrangian submanifolds 1053

2 Proof of Theorem 1.2

Suppose that L ↪→ T ∗Tn is an exact Lagrangian for some arbitrary n. Denote by p : L → Tn

the projection. Without restricting the generality we may suppose that p induces an epimor-phism at the level of fundamental groups: a result of Lalonde and Sikorav [16] asserts thatp∗(π1(L)) is of finite index in π1(Tn) = Zn . Consider the covering T ∗Tn → T ∗Tn whichcorresponds to this subgroup. We can lift the embedding of L to a Lagrangian embedding intothe covering space, (see [16] for the explicite formula). By construction, p∗ is an epimorphismfor the lifted submanifold. Now we prove the following:

Proposition 2.1 The subgroup K er(p∗) ⊂ π1(L) is finitely generated.

Proof According to 1.3 we have

(1) H∗(L; p∗u, Z/2) = 0

for any non-zero cohomology class u ∈ H1(Tn, R). In degree one the above relation is validfor integer coefficients: �Lemma 2.2 If L is closed and v ∈ H1(L , R) then

H1(L; v, Z/2) = 0 ⇐⇒ H1(L; v, Z) = 0.

Proof of the Lemma For v = 0 the statement is obvious. If v �= 0, choose α ∈ v withoutzeroes of index 0 and n (such a choice is always possible [17]). Consider the universal coverπ : ˜L → L and a primitive f̃ : ˜L → R of π∗α.

The proof is a straightforward consequence of a result of Sikorav [19] which asserts thatthe vanishing of H1(L; v) is equivalent to H0({ f̃ ≥ c}) = 0 for any real c, since the secondrelation is independent of the coefficient ring. �

The vanishing of H1(L; v, Z) is also equivalent to the fact that the class v belongs to theBieri-Neumann-Strebel invariant of the group π1(L). This invariant, denoted �1(G), definedfor a finitely generated group G, is an open subset of the sphere

S(G) = (Hom(G, R)\{0})/R∗+.

It was introduced in [4], where the above equivalence is shown. The remarquable fact aboutthis invariant is that it detects the finite generation of some subgroups of G. More precisely,we have the following theorem [4]:

Theorem 2.3 Let G be finitely generated and let N ⊂ G be a subgroup such that G/N isfree Abelian. Denote

S(G, N ) = {v ∈ S(G) | v|N = 0}.Then N is finitely generated if and only if S(G, N ) ⊂ �1(G).

Now we are able to finish the proof of 2.1. If v ∈ S(π1(L), K er(p∗)), then v = p∗u forsome u, so v ∈ �1(π1(L)) using the relation (1) and we can apply 2.3 to complete the proof.

�Let us now turn back to the case n = 3. We use a celebrated theorem of Stallings [20]:

Theorem 2.4 Let L be a closed manifold of dimension 3 and

v ∈ H1(L , Z) ≈ Hom(π1(L), Z).

If Ker(v) is finitely generated then there is a fibration f : L → S1 such that [ f ∗dθ ] = v.

123

1054 M. Damian

According to 2.1, any subgroup of π1(L) which contains K er(p∗) is finitely generated. Inparticular there is a fibration over the circle in any class of the form p∗u, for u ∈ H1(T3, Z).Let ui : Z3 → Z be the projections for i = 1, 2, 3 and denote vi = p∗ui . Choose a fibrationover the cercle in the class v1 and denote by F its fiber. We show that F is a torus.

We have π1(F) = Ker(v1). Consider the cohomology class of degree one on F definedby v′

2 = v2|π1(F). Obviously Ker(v′2) ⊃ Ker(p∗) so the former is finitely generated. Using

the “only if” part of 2.3 we get that the Novikov homology H1(F; v′2) is zero, and therefore

H∗(F, v′2) = 0. In particular χ(F) = 0, so F is either a torus or a Klein bottle. But the map

(v2, v3)|π1(F) defines an epimorphism of π1(F) onto Z2 which does not exist for the Kleinbottle. So F is a torus and L is a mapping torus associated to a diffeomorphism : T2 → T2.

It is easy to compute the homology of L:

H1(L , Z) = Z ⊕ (

Z2/Im(∗ − Id))

.

Since the projection p : L → T3 induces an epimorphism in H1 we infer that ∗ = Id. Wemay then apply a theorem of Epstein [10] which asserts that is isotopic to the identity. Inconclusion L is diffeormorphic to T3, as claimed.

In order to complete the proof of 1.2 we explain now why Th. 1.3 is valid for arbitrarycohomology classes u (recall that the proof in [6] is done for integer cohomology classesonly). The hypothesis u ∈ H1(M, Z) is used in the proof of the compactness propertyTh. 3.15 in [6]. A lower bound on the energy of holomorphic strips is needed in order toguarantee that the limit of a sequence of such strips is a broken trajectory built with a finitenumber of holomorphic strips. More precisely, one has to prove the following lemma:

Lemma 2.5 Let M be a closed manifold and let L ⊂ T ∗M be an exact Lagrangian subman-ifold. Consider a symplectic isotopy (φt ) and denote Lt = φt (L). Choose an almost complexstructure J on T ∗M which is compatible with the symplectic form and define the space ofJ -holomorphic strips

M = {

v ∈ C∞(R × [0, 1], T ∗M) | ∂̄J v = 0, v(s, 0) ∈ L0, v(s, 1) ∈ L1, E(v) < +∞}

where the energy E is given by

E(v) =∫

R×[0,1]

∣

∣

∣

∣

∣

∣

∣

∣

∂v

∂s

∣

∣

∣

∣

∣

∣

∣

∣

dsdt,

the norm being defined by the metric associated to J .Then there exists c > 0 such that for any non-constant v ∈ M

E(v) > c.

Proof Suppose the contrary: there is a sequence of non-constant elements (vn) in M suchthat

limn→+∞ E(vn) = 0.

For x, y ∈ L0 ∩ L1 denote

M(x, y) ={

v ∈ M | lims→−∞ v(s, ·) = x, lim

s→+∞ v(s, ·) = y

}

.

Since

M =⋃

x,y∈L0∩L1

M(x, y)

123

Floer–Novikov homology and applications to Lagrangian submanifolds 1055

by the work of Floer ([11], Prop. 1.b) and Oh ([18], Prop. 3.2), we may suppose that vn

belongs to some M(x, y) for any n. Proceeding like in the proof of Th. 3.15 in [6] we obtainthat there exists a real sequence (σn) such that v′

n(s, t) := (vn(s + σn, t)) converges in thetopology C∞

loc towards some non-constant limit v ∈ M(x, z), where z ∈ L0 ∩ L1 (note thatz = x is possible here).

On the other hand, since E(v′n) = E(vn), the energy of v must vanish, i.e. v is constant,

which is contradictory. �The proof of Theorem 1.2 is now complete.

3 Proof of Theorem 1.4

Let L̂ be the obvious one-point compactification of L . Due to (the proof of) the Poincaréconjecture and to the theorem of h-cobordism it is sufficient to show the following:

Proposition 3.1 Under the hypothesis of 1.4 the compactification L̂ is a simply connectedhomology sphere.

We start by noticing the following obvious fact:

Remark 3.2 If L is like in the statement of 1.4, then for any compact manifold M there is anexact Lagrangian embedding of the connected sum L̂#M into T ∗M .

Indeed, one just has to push the Lagrangian L by the flow of the radial vector field, so thatit becomes standard out of D × Rn for a small disk D ⊂ 0Rn , then cut a small disk D ⊂ 0M

and replace it by L ∩ (D × Rn) in D × Rn ⊂ T ∗M . The restriction to L of the Liouvilleform

∑

pi dqi is the differential d f of some function f : L → R which is constant outsideD. In particular the integral of the Liouville form on T ∗M vanishes on every loop of the newsubmanifold L̂#M , and therefore this submanifold is exact Lagrangian.

The next step of the proof is the following result (see [6], Corollary 1.2.b):

Proposition 3.3 Suppose n ≥ 3 and let Qn−1 be a closed connected manifold.

(a) Suppose that the connected sum L1#L2 of two closed manifolds admits an exactLagrangian embedding into T ∗(Q × S1). Then one of the Li is a simply connectedZ/2-homology sphere.

(b) Suppose moreover that n ≥ 4, that both Li are orientable and satisfy H2(Li , Z/2) = 0.Then one of the Li is a simply connected Z-homology sphere.

In order to get the proof of Theorem 1.4 from the previous proposition (via 3.1), one takesfor instance M = S1 × Sn−1 in 3.2 above. Applying the point (a) of 3.3 we get that L̂ issimply connected. For n = 3 this implies 3.1, by Poincaré duality. For n ≥ 4 we can applypoint (b) of the previous statement, whose hypothesis is fulfilled by point (a). This impliesagain Prop. 3.1 and thus the proof is complete.

So, the only thing to prove is Proposition 3.3. The point (a) was proved in [6], as acorollary of Theorem 1.3 (Corollary 1.2.b in [6]). To prove (b) we need the vanishing ofthe Novikov homology with integer coefficients in Th. 1.3. Therefore we have to defineFloer–Novikov homology with integer coefficients. This can be done if we provide coherentorientations on the spaces M(x, y). As shown in [12], chap. VIII, such orientations exist ifthe Lagrangian submanifold L is spin (or more generally relatively spin, meaning that the sec-ond Stiefel–Whitney class w2(L) lies in the image of the map H2(M, Z/2) → H2(L , Z/2)

123

1056 M. Damian

for a Lagrangian L ⊂ M). In particular, this is the case when L is orientable and satisfiesH2(L , Z/2) = 0, which is the hypothesis of (b).

Starting form the version with integer coefficients of Th. 1.3, one repeats the proof of thecorollary 1.2.b in [6], to get the desired conclusion.

References

1. Abouzaid, M.: On the homotopy type of nearby Lagrangians, arXiv:1005.0358v1 [math.SG] (2010)2. Arnol’d, V.I.: First steps in symplectic topology. Russ. Math. Surv. 6, 3–18 (1986)3. Audin, M.: Fibrés normaux d’immersions en dimension double, points doubles d’immersions lagrangi-

ennes et plongements totalement réels. Comment. Math. Helv. 63, 593–623 (1988)4. Bieri, R., Neumann, W.D., Strebel, R.: A geometric invariant for discrete groups. Invent. Math. 90, 451–

477 (1987)5. Damian, M.: Formes fermées non singulières et propriétés de finitude des groupes. Ann. Sci. Ec. Norm.

Sup. 33(3), 301–320 (2000)6. Damian, M.: Constraints on exact Lagrangians in cotangent bundles of manifolds fibered over the circle.

Comment. Math. Helv. 89, 705–746 (2009)7. Duff, D.: Symplectic manifolds with contact type boundaries. Invent. Math. 103(3), 651–671 (1991)8. Eliashberg, Y.: On symplectic manifolds with some contact properties. J. Differ. Geom 33(1), 233–238

(1991)9. Eliashberg, Y., Polterovich, L.: Local Lagrangian 2-knots are trivial. Ann. Math. 44, 61–76 (1996)

10. Epstein, D.: Curves on 2-manifolds and isotopies. Acta Math. 115, 83–107 (1966)11. Floer, A.: Symplectic fixed points and holomorphic spheres. Commun. Math. Phys. 120(4), 575–611

(1989)12. Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian Intersection Floer Theory. Anomaly and Obstruc-

tion. Part II. Studies in Adv. Math., American Math. Soc, International Press, Providence (2009)13. Fukaya, K., Seidel, P., Smith, I.: The symplectic geometry of cotangent bundles from categorical view-

point. Homological mirror symmetry. Lecture Notes in Phys., vol. 757, pp. 1–26. Springer, Berlin (2009)14. Gromov, M.: Pseudo-holomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985)15. Hind, R.: Lagrangian spheres in S2 × S2. Geom. Funct. Anal. 14(2), 303–318 (2004)16. Lalonde, F., Sikorav, J.-C.: Sous-variétés lagrangiennes et lagrangiennes exactes des fibrés cotangents.

Comment. Math. Helv. 66, 18–33 (1991)17. Levitt, G.: Formes fermées singulières et groupe fondamental. Invent. Math. 88(3), 635–667 (1987)18. Oh, Y.-G.: Floer cohomology of Lagrangian intersections and pseudoholomorphic disks. I. Commun. Pure

Appl. Math. 46, 949–993 (1993)19. Sikorav, J.-C.: Homologie de Novikov associée à une classe de cohomologie réelle de degré un. Thèse

Orsay (1987)20. Stallings, J.R.: On fibering certains 3-manifolds. Topology of 3-manifolds and related topics, pp. 95–100.

Prentice-Hall, Englewood Cliffs (1962)21. Viterbo, C.: Functors and computations in Floer homology with applications. I. Geom. Funct. Anal. 9(5),

985–1033 (1999)

123