The Role of the Legendre Transform in the Study of the Floer Complex of Cotangent Bundles

Consider a classical Hamiltonian H on the cotangent bundle T*M of a closed orientable manifold M, and let L:TM → R be its Legendre‐dual Lagrangian. In a previous paper we constructed an isomorphism Φ from the Morse complex of the Lagrangian action functional that is associated to L to the Floer complex that is determined by H. In this paper we give an explicit construction of a homotopy inverse Ψ of Φ. Contrary to other previously defined maps going in the same direction, Ψ is an isomorphism at the chain level and preserves the action filtration. Its definition is based on counting Floer trajectories on the negative half‐cylinder that on the boundary satisfy half of the Hamilton equations. Albeit not of Lagrangian type, such a boundary condition defines Fredholm operators with good compactness properties. We also present a heuristic argument which, independently of any Fredholm and compactness analysis, explains why the spaces of maps that are used in the definition of Φ and Ψ are the natural ones. The Legendre transform plays a crucial role both in our rigorous and in our heuristic arguments. We treat with some detail the delicate issue of orientations and show that the homology of the Floer complex is isomorphic to the singular homology of the loop space of M with a system of local coefficients, which is defined by the pullback of the second Stiefel‐Whitney class of TM on 2‐tori in M.© 2015 Wiley Periodicals, Inc.     

Floer Homology and the Heat Flow

We study the heat flow in the loop space of a closed Riemannian manifold M as an adiabatic limit of the Floer equations in the cotangent bundle. Our main application is a proof that the Floer homology of the cotangent bundle, for the Hamiltonian function kinetic plus potential energy, is naturally isomorphic to the homology of the loop space. J.W. received partial financial support from TH-Projekt 00321. Received: December 2004 Revision: September 2005 Accepted: September 2005     

L 2-Topology and Lagrangians in the Space of Connections Over a Riemann Surface

We examine the L ²-topology of the gauge orbits over a closed Riemann surface. We prove a subtle local slice theorem based on the div-curl lemma of harmonic analysis, and deduce local pathwise connectedness of the gauge orbits. Based on a quantitative version of the connectivity, we generalize compactness results for anti-self-dual instantons with Lagrangian boundary conditions to general gauge-invariant Lagrangian submanifolds. This provides the foundation for the construction of instanton Floer homology for pairs of a 3-manifold with boundary and a Lagrangian in the configuration space over the boundary.     

Localization for Involutions in Floer Cohomology

We consider Lagrangian Floer cohomology for a pair of Lagrangian submanifolds in a symplectic manifold M. Suppose that M carries a symplectic involution, which preserves both submanifolds. Under various topological hypotheses, we prove a localization theorem for Floer cohomology, which implies a Smith-type inequality for the Floer cohomology groups in M and its fixed point set. Two applications to symplectic Khovanov cohomology are included.     

Corrigendum: On the Floer Homology of Cotangent Bundles

We fix an orientation issue that appears in our previous paper about the isomorphism between Floer homology of cotangent bundles and loop space homology. When the second Stiefel‐Whitney class of the underlying manifold does not vanish on 2‐tori, this isomorphism requires the use of a twisted version of the Floer complex. © 2014 Wiley Periodicals, Inc.     

Floer homology for magnetic fields with at most linear growth on the universal cover

The Floer homology of a cotangent bundle is isomorphic to loop space homology of the underlying manifold, as proved by Abbondandolo and Schwarz, Salamon and Weber, and Viterbo. In this paper we show that in the presence of a Dirac magnetic monopole which admits a primitive with at most linear growth on the universal cover, the Floer homology in atoroidal free homotopy classes is again isomorphic to loop space homology. As a consequence we prove that for any atoroidal free homotopy class and any sufficiently small $\tau >0$, any magnetic flow associated to the Dirac magnetic monopole has a closed orbit of period τ belonging to the given free homotopy class. In the case where the Dirac magnetic monopole admits a bounded primitive on the universal cover we also prove the Conley conjecture for Hamiltonians that are quadratic at infinity, i.e., we show that such Hamiltonians have infinitely many periodic orbits.     

Lagrangian Rabinowitz Floer homology and twisted cotangent bundles

We study the following rigidity problem in symplectic geometry: can one displace a Lagrangian submanifold from a hypersurface? We relate this to the Arnold Chord Conjecture, and introduce a refined question about the existence of relative leaf-wise intersection points, which are the Lagrangian-theoretic analogue of the notion of leaf-wise intersection points defined by Moser (Acta. Math. 141(1–2):17–34, 1978). Our tool is Lagrangian Rabinowitz Floer homology, which we define first for Liouville domains and exact Lagrangian submanifolds with Legendrian boundary. We then extend this to the ‘virtually contact’ setting. By means of an Abbondandolo–Schwarz short exact sequence we compute the Lagrangian Rabinowitz Floer homology of certain regular level sets of Tonelli Hamiltonians of sufficiently high energy in twisted cotangent bundles, where the Lagrangians are conormal bundles. We deduce that in this situation a generic Hamiltonian diffeomorphism has infinitely many relative leaf-wise intersection points.     

Hamiltonian dynamics on convex symplectic manifolds

We study the dynamics of Hamiltonian diffeomorphisms on convex symplectic manifolds. To this end we first establish an explicit isomorphism between the Floer homology and the Morse homology of such a manifold, and then use this isomorphism to construct a biinvariant metric on the group of compactly supported Hamiltonian diffeomorphisms analogous to the metrics constructed by Viterbo, Schwarz and Oh. These tools are then applied to prove and reprove results in Hamiltonian dynamics. Our applications comprise a uniform lower estimate for the slow entropy of a compactly supported Hamiltonian diffeomorphism, the existence of infinitely many non-trivial periodic points of a compactly supported Hamiltonian diffeomorphism of a subcritical Stein manifold, new cases of the Weinstein conjecture, and, most noteworthy, new existence results for closed trajectories of a charge in a magnetic field on almost all small energy levels. We shall also obtain some new Lagrangian intersection results. Partially supported by the Swiss National Foundation. Supported by the Swiss National Foundation and the von Roll Research Foundation.     

Quantum -modules and generalized mirror transformations

In the previous paper [Hiroshi Iritani, Quantum D-modules and equivariant Floer theory for free loop spaces, Math. Z. 252 (3) (2006) 577–622], the author defined equivariant Floer cohomology for a complete intersection in a toric variety and showed that it is isomorphic to the small quantum D-module after a mirror transformation when the first Chern class c₁(M) of the tangent bundle is nef. In this paper, even when c₁(M) is not nef, we show that the equivariant Floer cohomology reconstructs the big quantum D-module under certain conditions on the ambient toric variety. The proof is based on a mirror theorem of Coates and Givental [T. Coates, A.B. Givental, Quantum Riemann — Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (1) (2007) 15–53]. The reconstruction procedure here gives a generalized mirror transformation first observed by Jinzenji in low degrees [Masao Jinzenji, On the quantum cohomology rings of general type projective hypersurfaces and generalized mirror transformation, Internat. J. Modern Phys. A 15 (11) (2000) 1557–1595; Masao Jinzenji, Co-ordinate change of Gauss–Manin system and generalized mirror transformation, Internat. J. Modern Phys. A 20 (10) (2005) 2131–2156].     

Morse theory for periodic solutions of hamiltonian systems and the maslov index

In this paper we prove Morse type inequalities for the contractible 1-periodic solutions of time dependent Hamiltonian differential equations on those compact symplectic manifolds M for which the symplectic form and the first Chern class of the tangent bundle vanish over π₂ (M). The proof is based on a version of infinite dimensional Morse theory which is due to Floer. The key point is an index theorem for the Fredholm operator which plays a central role in Floer homology. The index formula involves the Maslov index of nondegenerate contractible periodic solutions. This Maslov index plays the same role as the Morse index of a nondegenerate critical point does in finite dimensional Morse theory. We shall use this connection between Floer homology and Maslov index to establish the existence of infinitely many periodic solutions having integer periods provided that every 1-periodic solution has at least one Floquet multiplier which is not equal to 1.     

Morse homology for the heat flow

We use the heat flow on the loop space of a closed Riemannian manifold—viewed as a parabolic boundary value problem for infinite cylinders—to construct an algebraic chain complex. The chain groups are generated by perturbed closed geodesics. The boundary operator is defined by counting, modulo time shift, heat flow trajectories between geodesics of Morse index difference one. By Salamon and Weber (GAFA 16:1050–138, 2006) this heat flow homology is naturally isomorphic to Floer homology of the cotangent bundle for Hamiltonians given by kinetic plus potential energy.     

Coherent orientation of mixed moduli spaces in Morse-Floer theory

We give the coherent orientation for the spaces of intersections of gradient trajectories and holomorphic disks in cotangent bundle. This construction provides the Piunikhin-Salamon-Schwarz isomorphism between Morse homology and Floer homology for Lagrangian intersections in cotangent bundles, with integer coefficients. This work is partially supported by Ministry of Science and Environmental Protection of Republic of Serbia Project #144020.     

Homological stability for the mapping class groups of non-orientable surfaces

We prove that the homology of the mapping class groups of non-orientable surfaces stabilizes with the genus of the surface. Combining our result with recent work of Madsen and Weiss, we obtain that the classifying space of the stable mapping class group of non-orientable surfaces, up to homology isomorphism, is the infinite loop space of a Thom spectrum built from the canonical bundle over the Grassmannians of 2-planes in ℝ ⁿ⁺². In particular, we show that the stable rational cohomology is a polynomial algebra on generators in degrees 4i – this is the non-oriented analogue of the Mumford conjecture.     

Morse homology for the heat flow – Linear theory

Consider the linear parabolic partial differential equation ${\mathcal {D}}_u\xi =0$ which arises by linearizing the heat flow on the loop space of a Riemannian manifold M. The solutions are vector fields along infinite cylinders u in M. For these solutions we establish regularity and a priori estimates. We show that for nondegenerate asymptotic boundary conditions the solutions decay exponentially in L² in forward and backward time. In this case ${\mathcal {D}}_u$ viewed as linear operator from the parabolic Sobolev space ${\mathcal {W}}^{1,p}$ to L^p is Fredholm whenever p > 1. We close with an L^p estimate for products of first order terms which is a crucial ingredient in the sequel 13 to prove regularity and the implicit function theorem. The results of the present text are the base to construct in 13 an algebraic chain complex whose homology represents the homology of the loop space.     

A homotopy theoretic realization of string topology

Let M be a closed, oriented manifold of dimension d. Let LM be the space of smooth loops in M. In [2] Chas and Sullivan defined a product on the homology H _* (LM) of degree -d. They then investigated other structure that this product induces, including a Batalin -Vilkovisky structure, and a Lie algebra structure on the S¹ equivariant homology H _* ^{S 1} (LM). These algebraic structures, as well as others, came under the general heading of the ”string topology” of M. In this paper we will describe a realization of the Chas-Sullivan loop product in terms of a ring spectrum structure on the Thom spectrum of a certain virtual bundle over the loop space. We also show that an operad action on the homology of the loop space discovered by Voronov has a homotopy theoretic realization on the level of Thom spectra. This is the ” cactus operad” defined in [6] which is equivalent to operad of framed disks in . This operad action realizes the Chas - Sullivan BV structure on H _* (LM). We then describe a cosimplicial model of this ring spectrum, and by applying the singular cochain functor to this cosimplicial spectrum we show that this ring structure can be interpreted as the cup product in the Hochschild cohomology, HH ^* (C ^* (M); C ^* (M)). Received: 31 July 2001 / Revised version: 11 September 2001 Published online: 5 September 2002     

Three dimensional expansive diffeomorphisms with homoclinic points

LetM be a compact connected oriented three dimensional manifold andf:MM an expansive diffeomorphism such that (f)=M. Let us also assume that there is a hyperbolic periodic point with a homoclinic intersection. Thenf is conjugate to an Anosov isomorphism ofT ³. Moreover, we show that at a homoclinic point the stable and unstable manifolds of the hyperbolic periodic point are topologically transverse.     

An iterated graph construction and periodic orbits of Hamiltonian delay equations

According to the Arnold conjectures and Floer's proofs, there are non-trivial lower bounds for the number of periodic solutions of Hamiltonian differential equations on a closed symplectic manifold whose symplectic form vanishes on spheres. We use an iterated graph construction and Lagrangian Floer homology to show that these lower bounds also hold for certain Hamiltonian delay equations.     

Decomposable representations and Lagrangian submanifolds of moduli spaces associated to surface groups

The importance of explicit examples of Lagrangian submanifolds of moduli spaces is revealed by papers such as Dostoglou and Salamon (Ann. of Math (2), 139(3), 581–640, 1994) and Salamon (Proceedings of the international congress of mathematicians, vol.1, 2 (Zürich, 1994), pp. 526–536. Birkhäuser, Basel, 1995): given a 3-manifold M with boundary ?M = Σ, Dostoglou and Salamon use such examples to obtain a proof of the Atiyah-Floer conjecture relating the symplectic Floer homology of the representation space Hom(π₁(Σ = ?M), U)/U (associated to an explicit pair of Lagrangian submanifolds of this representation space) and the instanton homology of the 3-manifold M. In the present paper, we construct a Lagrangian submanifold of the space of representations ${\mathcal{M}_{g,l}:=Hom_\mathcal{C}(\pi_{g,l}, U)/U}The importance of explicit examples of Lagrangian submanifolds of moduli spaces is revealed by papers such as Dostoglou and Salamon (Ann. of Math (2), 139(3), 581–640, 1994) and Salamon (Proceedings of the international congress of mathematicians, vol.1, 2 (Zürich, 1994), pp. 526–536. Birkh?user, Basel, 1995): given a 3-manifold M with boundary ∂M = Σ, Dostoglou and Salamon use such examples to obtain a proof of the Atiyah-Floer conjecture relating the symplectic Floer homology of the representation space Hom(π₁(Σ = ∂M), U)/U (associated to an explicit pair of Lagrangian submanifolds of this representation space) and the instanton homology of the 3-manifold M. In the present paper, we construct a Lagrangian submanifold of the space of representations of the fundamental group π _g,l of a punctured Riemann surface Σ _g,l into an arbitrary compact connected Lie group U. This Lagrangian submanifold is obtained as the fixed-point set of an anti-symplectic involution defined on . We show that the involution is induced by a form-reversing involution β defined on the quasi-Hamiltonian space . The fact that has a non-empty fixed-point set is a consequence of the real convexity theorem for group-valued momentum maps proved in Schaffhauser (A real convexity theorem for quasi-Hamiltonian actions, submitted, 25 p, 2007. ). The notion of decomposable representation provides a geometric interpretation of the Lagrangian submanifold thus obtained. Supported by the Japanese Society for Promotion of Science (JSPS).     

Isomorphic chain complexes of Hamiltonian dynamics on tori

In this work, for a given smooth, generic Hamiltonian ${H : \mathbb{S}^{1} \times \mathbb{T}^{2n} \rightarrow \mathbb{R}}$ on the torus ${\mathbb{T}^{2n} = \mathbb{R}^{2n}/\mathbb{Z}^{2n}}$ we construct a chain isomorphism ${\Phi_{*} : (C_{*}(H), \partial^{M}_{*}) \rightarrow (C_{*}(H), \partial^{F}_{*})}$ between the Morse complex of the Hamiltonian action AH on the free loop space of the torus ${\Lambda_{0}(\mathbb{T}^{2n})}$ and the Floer complex. Though both complexes are generated by the critical points of A _H , their boundary operators differ. Therefore, the construction of ${\Phi}$ is based on counting the moduli spaces of hybrid-type solutions which involves stating a new non-Lagrangian boundary value problem for Cauchy–Riemann type operators not yet studied in Floer theory. We finally want to note that the problem is completely symmetric. So we also could construct an isomorphism ${\Psi_{*} : (C_{*}(H), \partial^{F}_{*}) \rightarrow (C_{*}(H), \partial^{M}_{*})}$ .     

Hofer’s metric on the space of Lagrangian submanifolds and wrapped Floer homology

In this paper, we study the relationship between wrapped Floer homology and displaceability of a Lagrangian submanifold which we call vanishing theorem of wrapped Floer homology. We also use this theorem to study Hofer’s pseudometric on the space of Lagrangian submanifolds. We prove an inequality, the Lagrangian version of the inequality of Gromov width and displacement energy, which is called energy-capacity inequality.