On the Floer homology of cotangent bundles

This paper concerns Floer homology for periodic orbits and for a Lagrangian intersection problem on the cotangent bundle T* M of a compact orientable manifold M. The first result is a new L^∞ estimate for the solutions of the Floer equation, which allows us to deal with a larger—and more natural—class of Hamiltonians. The second and main result is a new construction of the isomorphism between the Floer homology and the singular homology of the free loop space of M in the periodic case, or of the based loop space of M in the Lagrangian intersection problem. The idea for the construction of such an isomorphism is to consider a Hamiltonian that is the Legendre transform of a Lagrangian on T M and to construct an isomorphism between the Floer complex and the Morse complex of the classical Lagrangian action functional on the space of W^1,2 free or based loops on M. © 2005 Wiley Periodicals, Inc.     

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.     

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.     

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.     

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.     

Angle theorems for the Lagrangian mean curvature flow

We prove that symplectic maps between Riemann surfaces L, M of constant, nonpositive and equal curvature converge to minimal symplectic maps, if the Lagrangian angle for the corresponding Lagrangian submanifold in the cross product space satisfies . If one considers a 4-dimensional K?hler-Einstein manifold of nonpositive scalar curvature that admits two complex structures J, K which commute and assumes that is a compact oriented Lagrangian submanifold w.r.t. J such that the K?hler form w.r.t.K restricted to L is positive and , then L converges under the mean curvature flow to a minimal Lagrangian submanifold which is calibrated w.r.t. . Received: 11 April 2001 / Published online: 29 April 2002     

The existence of solutions of a general boundary value problem for the divergence

This paper is concerned with the question of necessary and sufficient conditions to find a vector field V∈ Γ(TM) solving the equation div V = Φ under inhomogeneous boundary conditions V|_?M = Z|_?M with Z∈ Γ (TM) An existence and regularity result is given for an arbitrary Riemannian manifold with boundary, M. The proof is based on the Hodge theory of differential forms.     

The Künneth formula in Floer homology for manifolds with restricted contact type boundary

We prove the Künneth formula in Floer (co)homology for manifolds with restricted contact type boundary. We use Viterbo's definition of Floer homology, involving the symplectic completion by adding a positive cone over the boundary. The Künneth formula implies the vanishing of Floer (co)homology for subcritical Stein manifolds. Other applications include the Weinstein conjecture in certain product manifolds, obstructions to exact Lagrangian embeddings, existence of holomorphic curves with Lagrangian boundary condition, as well as symplectic capacities. Supported by ENS Lyon, école Polytechnique (Palaiseau) and ETH (Zürich).     

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     

Functors and Computations in Floer Homology with Applications, I

This paper is concerned with Floer cohomology of manifolds with contact type boundary. In this case, there is no conjecture on this ring, as opposed to the compact case, where it is isomorphic to the usual cohomology (with the quantum product). We construct two mappings in Floer cohomology and prove some functorial properties of these two mappings. The first one is a map from the Floer cohomology of M to the relative cohomology of M modulo its boundary. The other is associated to a codimension zero embedding, and may be considered as a cohomological transfer. These maps are used to define some properties of symplectic manifolds with contact type boundary. These are algebraic versions of the Weinstein conjecture, asserting existence of closed characteristics on . This is proved for many cases, Euclidean space and subcritical Stein manifolds, vector bundles, products, cotangent bundles. It is also proved that the above property implies some restrictions on Lagrangian embeddings, and also yields in certain cases, existence results for holomorphic curves bounded by the Lagrange submanifold. The last section is devoted to applications of this existence result, to real forms of Stein manifolds and obstructions to polynomial convexity in Stein manifolds. Many of our applications rely on the computation of the Floer cohomology of a cotangent bundle, that is the subject of Part II. Submitted: December 1997, revised version: February 1999.     

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.     

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 link surgery spectral sequence in monopole Floer homology

To a link L⊂S³, we associate a spectral sequence whose E² page is the reduced Khovanov homology of L and which converges to a version of the monopole Floer homology of the branched double cover. The pages Ek for k?2 depend only on the mutation equivalence class of L. We define a mod 2 grading on the spectral sequence which interpolates between the δ-grading on Khovanov homology and the mod 2 grading on Floer homology. We also derive a new formula for link signature that is well adapted to Khovanov homology.More generally, we construct new bigraded invariants of a framed link in a 3-manifold as the pages of a spectral sequence modeled on the surgery exact triangle. The differentials count monopoles over families of metrics parameterized by permutohedra. We utilize a connection between the topology of link surgeries and the combinatorics of graph-associahedra. This also yields simple realizations of permutohedra and associahedra, as refinements of hypercubes.     

Infima of universal energy functionals on homotopy classes

We consider the infima (f) on homotopy classes of energy functionals E defined on smooth maps f: Mⁿ → V^k between compact connected Riemannian manifolds. If M contains a sub‐manifold L of codimension greater than the degree of E then (f) is determined by the homotopy class of the restriction of f to M \ L. Conversely if the infimum on a homotopy class of a functional of at least conformal degree vanishes then the map is trivial in homology of high degrees. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rigidity and gluing for Morse and Novikov complexes

We obtain rigidity and gluing results for the Morse complex of a real-valued Morse function as well as for the Novikov complex of a circle-valued Morse function. A rigidity result is also proved for the Floer complex of a hamiltonian defined on a closed symplectic manifold (M,) with c₁|_2(M)=[]|_2(M)=0. The rigidity results for these complexes show that the complex of a fixed generic function/hamiltonian is a retract of the Morse (respectively Novikov or Floer) complex of any other sufficiently C⁰ close generic function/hamiltonian. The gluing result is a type of Mayer-Vietoris formula for the Morse complex. It is used to express algebraically the Novikov complex up to isomorphism in terms of the Morse complex of a fundamental domain. Morse cobordisms are used to compare various Morse-type complexes without the need of bifurcation theory.     

Lagrangian Floer theory on compact toric manifolds II: bulk deformations

This is a continuation of part I in the series of the papers on Lagrangian Floer theory on toric manifolds. Using the deformations of Floer cohomology by the ambient cycles, which we call bulk deformations, we find a continuum of non-displaceable Lagrangian fibers on some compact toric manifolds. We also provide a method of finding all fibers with non-vanishing Floer cohomology with bulk deformations in arbitrary compact toric manifolds, which we call bulk-balanced Lagrangian fibers.     

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).     

A General Fredholm Theory II: Implicit Function Theorems

This is the second paper in a series introducing a generalized Fredholm theory in a new class of smooth spaces called polyfolds. In general, these spaces are not locally homeomorphic to open sets in Banach spaces. The current paper develops the Fredholm theory in M-polyfold bundles. It consists of a transversality and a perturbation theory. In upcoming papers the generalized Fredholm theory will be applied to the Floer Theory, the Gromov–Witten Theory and the Symplectic Field Theory H.H.’s research partially supported by NSF grant DMS-0603957. K.W.’s research partially supported by NSF grant DMS-0606588.     

A superquadratic indefinite elliptic system and its Morse–Conley–Floer homology

We study critical points of the indefinite functional by applying Floer's homology construction to the ordinary gradient flow of the functional f on a suitable Sobolev space. One of our main observations is that even though this flow is well posed in both time directions and lacks any kind of smoothing property one can still obtain compactness of connecting orbit spaces and thus define the Floer homology for . Received November 11, 1997; in final form March 12, 1998     

Moduli spaces of vector bundles over a Klein surface

A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface M endowed with an anti-holomorphic involution which determines topologically the original surface S. In this paper, we compare dianalytic vector bundles over S and holomorphic vector bundles over M, devoting special attention to the implications that this has for moduli varieties of semistable vector bundles over M. We construct, starting from S, totally real, totally geodesic, Lagrangian submanifolds of moduli varieties of semistable vector bundles of fixed rank and degree over M. This relates the present work to the constructions of Ho and Liu over non-orientable compact surfaces with empty boundary (Ho and Liu in Commun Anal Geom 16(3):617–679, 2008).