Symplectic topology and floer homology: Volume 2: Floer homology and its applications
Abstract
Published in two volumes, this is the first book to provide a thorough and systematic explanation of symplectic topology, and the analytical details and techniques used in applying the machinery arising from Floer theory as a whole. Volume 2 provides a comprehensive introduction to both Hamiltonian Floer theory and Lagrangian Floer theory, including many examples of their applications to various problems in symplectic topology. The first volume covered the basic materials of Hamiltonian dynamics and symplectic geometry and the analytic foundations of Gromov's pseudoholomorphic curve theory. Symplectic Topology and Floer Homology is a comprehensive resource suitable for experts and newcomers alike.
This is the first of a series of two articles where we construct a version of wrapped Fukaya category of the cotangent bundle of the knot complement of a compact 3-manifold M, and do some calculation for the case of hyperbolic knots . For the construction, we use the wrapping induced by the kinetic energy Hamiltonian associated to the cylindrical adjustment on of a smooth metric g defined on M. We then consider the torus as an object in this category and its wrapped Floer complex where N(K) is a tubular neighborhood of . We prove that the quasi-equivalence class of the category and the quasi-isomorphism class of the algebra are independent of the choice of cylindrical adjustments of such metrics depending only on the isotopy class of the knot K in M. In a sequel (Bae et al. in Asian J Math 25(1):117–176, 2019), we give constructions of a wrapped Fukaya category for hyperbolic knot K and of algebra directly using the hyperbolic metric h on , and prove a formality result for the asymptotic boundary of .
Local normal form theorems for smooth equivariant maps between infinite-dimensional manifolds are established. These normal form results are new even in finite dimensions. The proof is inspired by the Lyapunov–Schmidt reduction for dynamical systems and by the Kuranishi method for moduli spaces. It uses a slice theorem for Fréchet manifolds as the main technical tool. As a consequence, the abstract moduli space obtained by factorizing a level set of the equivariant map with respect to the group action carries the structure of a Kuranishi space, i.e., such moduli spaces are locally modeled on the quotient by a compact group of the zero set of a smooth map. The general results are applied to the moduli space of anti-self-dual instantons, the Seiberg–Witten moduli space and the moduli space of pseudoholomorphic curves.
In this paper, we develop a mini-max theory of the action functional over the semi-infinite cycles via the chain level Floer homology theory and construct spectral invariants of Hamiltonian paths on arbitrary, especially on nonexact and nonrational, compact symplectic manifold (M, ω). To each given time dependent Hamiltonian function H and quantum cohomology class 0 ≠ a ∈ QH*(M), we associate an invariant ρ(H; a) which varies continuously over H in the C
0-topology. This is obtained as the mini-max value over the semi-infinite cycles whose homology class is “dual” to the given quantum cohomology class a on the covering space \(
\tilde \Omega _0 (M)
\) of the contractible loop space Ω0(M). We call them the Novikov Floer cycles. We apply the spectral invariants to the study of Hamiltonian diffeomorphisms in sequels of this paper.
We assume that (M, ω) is strongly semipositive here, to be removed in a sequel to this paper.
In this paper we provide a criterion for the quasi-autonomous Hamiltonian path (``Hofer's geodesic'') on arbitrary closed symplectic manifolds to be length minimizing in its homotopy class in terms of the spectral invariants that the author has recently constructed (math.SG/0206092). As an application, we prove that any autonomous Hamiltonian path on arbitrary closed symplectic manifolds is length minimizing in {\it its homotopy class} with fixed ends, when it has no contractible periodic orbits {\it of period one}, has a maximum and a minimum point which are generically under-twisted and all of its critical points are nondegenerate in the Floer theoretic sense. This is a sequel to the papers math.SG/0104243 and math.SG/0206092.
We give a complete proof for Gromov's compactness theorem for pseudo holomorphic curves both in the case of closed curves and curves with boundary.
The main purpose of this paper is to carry out some of the foundational study of -Hamiltonian geometry and -symplectic topology. We introduce the notions of the strong and the weak {\it Hamiltonian topology} on the space of Hamiltonian paths, and on the group of Hamiltonian diffeomorphisms. We then define the {\it group} and the space of {\it Hamiltonian homeomorphisms} such that where is the group of symplectic homeomorphisms. We prove that is a {\it normal subgroup} of and contains all the time-one maps of Hamiltonian vector fields of -functions. We prove that is path connected and so contained in the identity component of . In the case of an orientable surface, we prove that the {\it mass flow} of any element from vanishes, which in turn implies that is strictly smaller than the identity component of the group of area preserving homeomorphisms when . For the case of , we conjecture that is still a proper subgroup of .
Floer (co)homology of the symplectic manifold which was originally introduced by Floer in relation to the Arnol'd conjecture has recently attracted much attention from both mathematicians and mathematical physicists either from the algebraic point of view or in relation to the quantum cohomology and the mirror symmetry. Very recent progress in its relative version, the Floer (co)homology of Lagrangian submanifolds, has revealed quite different mathematical perspective: The Floer (co)homology theory of Lagrangian submanifolds is a powerful tool in the study of symplectic topology of Lagrangian submanifolds, just as the classical (co)homology theory in topology has been so in the study of differential topology of submanifolds on differentiable manifolds. In this survey, we will review the Floer theory of Lagrangian submanifolds and explain the recent progress made by Chekanov and by the present author in this relative Floer theory, which have found several applications of the Floer theory to the symplectic topology of Lagrangian submanifolds which demonstrates the above perspective: They include a Floer theoretic proof of Gromov's non-exactness theorem, an optimal lower bound for the symplectic disjunction energy, construction of new symplectic invariants, proofs of the non-degeneracy of Hofer's distance on the space of Lagrangian submanifolds in the cotangent bundles and new results on the Maslov class obstruction to the Lagrangian embedding in Cn. Each of these applications accompanies new development in the Floer theory itself: localizations, semi-infinite cycles and minimax theory in the Floer theory and a spectral sequence as quantum cor- rections and others. We will also define the relative version of the quantum cohomology, and explain its relation to the Floer cohomology of Lagrangian submanifolds and its applications.
In this paper, we study the structure ofJ-holomorphic discs in relation to the Fredholm theory of pseudo-holomorphic discs with totally real boundary conditions in
almost complex manifolds (M, J). We prove that anyJ-holomorphic disc with totally real boundary condition that is injective in the interior except at a discrete set of points,
which we call a “normalized disc,” must either have some boundary point that is regular and has multiplicity one, or satisfy
that its image forms a smooth immersed compact surface (without boundary) with a finite number of self-intersections and a
finite number of branch points. In the course of proving this theorem, we also prove several theorems on the local structure
of boundary points ofJ-holomorphic discs, and as an application we give a complete treatment of the transverslity result for Floer’s pseudo-holomorphic
trajectories for Lagrangian intersections in symplectic geometry.
We prove the following special case of the Arnold conjec ture on the fixed points of an exact deformation
These are lecture notes from my talks at the "Current Developments in Mathematics" conference (Harvard, 2006). They cover a variety of topics involving symplectic cohomology. In particular, a discussion of (algorithmic) classification issues in symplectic and contact topology is included. Comment: 37 pages, 2 figures; v2 has added references, slightly expanded discussion of algorithmic recognizability, more balanced account of early developments; v3 has clarifications in the discussion of equivariant symplectic cohomology; v4 with terminology change (Liouville instead of Weinstein), by popular request; v5, one nasty typo removed
The chain complexes underlying Floer homology theories typically carry a
real-valued filtration, allowing one to associate to each Floer homology class
a spectral number defined as the infimum of the filtration levels of chains
representing that class. These spectral numbers have been studied extensively
in the case of Hamiltonian Floer homology by Oh, Schwarz, and others. We prove
that the spectral number associated to any nonzero Floer homology class is
always finite, and that the infimum in the definition of the spectral number is
always attained. In the Hamiltonian case, this implies that what is known as
the ``nondegenerate spectrality'' axiom holds on all closed symplectic
manifolds. Our proofs are entirely algebraic and apply to any Floer-type theory
(including Novikov homology) satisfying certain standard formal properties. The
key ingredient is a theorem about the existence of best approximations of
arbitrary elements of finitely generated free modules over Novikov rings by
elements of prescribed submodules with respect to a certain family of
non-Archimedean metrics.