A survey of the topological properties of symplectomorphism groups

The scattering matrix and associated formulas in Hamiltonian mechanics

We survey the basic notions of scattering theory in Hamiltonian mechanics with a particular attention to the analogies with scattering theory in quantum mechanics. We discuss the scattering symplectomorphism, which is analogous to the scattering matrix. We prove identities which relate the Calabi invariant of the scattering symplectomorphism to the total time delay and the regularised phase space volume. These identities are analogous to the Birman-Krein formula and the Eisenbud-Wigner formula in quantum scattering theory.

Cohomological properties of ruled symplectic structures

This survey presents some recent results by the authors and Polterovich on the topological properties of ruled symplectic manifolds. The bundle M \to P \to B that is associated with a ruled manifold has the group of Hamiltonian symplectomorphisms of M as structure group if the base is simply connected. Thus information about Hamiltonian bundles gives stability results for ruled structures as well as obstructions to their existence.

The growth rate of Floer homology and symplectic zeta function

The main theme of this paper is to study for a symplectomorphism of a compact surface, the asymptotic invariant which is defined to be the growth rate of the sequence of the total dimensions of symplectic Floer homologies of the iterates of the symplectomorphism. We prove that the asymptotic invariant coincides with asymptotic Nielsen number and with asymptotic absolute Lefschetz number. We also show that the asymptotic invariant coincides with the largest dilatation of the pseudo-Anosov components of the symplectomorphism and its logarithm coincides with the topological entropy. This implies …

Topological properties of Hamiltonian circle actions

This paper studies Hamiltonian circle actions, i.e. circle subgroups of the group Ham(M,\om) of Hamiltonian symplectomorphisms of a closed symplectic manifold (M,\om). Our main tool is the Seidel representation of π_1(\Ham(M,\om)) in the units of the quantum homology ring. We show that if the weights of the action at the points at which the moment map is a maximum are sufficiently small then the circle represents a nonzero element of π_1(\Ham(M,\om)). Further, if the isotropy has order at most two and the circle contracts in \Ham(M,\om) then the homology of M is invariant under an involution. …

Symplectic configurations

We define a class of symplectic fibrations called symplectic configurations. They are natural generalization of Hamiltonian fibrations. Their geometric and topological properties are investigated. We are mainly concentrated on integral symplectic manifolds. We construct the classifyng space \B of symplectic integral configurations. The properties of the classifying map \B --> BSymp(M,w) are examined. The universal symplectic bundle over \B has a natural connection whose holonomy group is isomorphic to the enlarged Hamiltonian group recently defined by McDuff. The space \B is identified with th…

Kuranishi atlases with trivial isotropy - the 2013 state of affairs

Kuranishi structures were introduced to symplectic topology by Fukaya and Ono and recently refined by Joyce, in order to extract homological data from compactified moduli spaces of holomorphic maps in cases where geometric regularization approaches such as perturbations of the almost complex structure do not yield a smooth structure on the moduli space. We give a general survey of regularization techniques in symplectic topology, pointing to some general analytic issues, and discussing some specific topological issues of the Kuranishi approach. In the main body of the paper we provide an abstr…

A topological introduction to knot contact homology

This is a survey of knot contact homology, with an emphasis on topological, algebraic, and combinatorial aspects.

拓扑场论中的同调代数与模空间

Homological algebra and moduli spaces in topological field theories

该研究为综述类工作，梳理了辛几何与规范理论领域的各类Floer理论及彼此关联。

Floer theory and its topological applications

We survey the different versions of Floer homology that can be associated to three-manifolds. We also discuss their applications, particularly to questions about surgery, homology cobordism, and four-manifolds with boundary. We then describe Floer stable homotopy types, the related Pin(2)-equivariant Seiberg-Witten Floer homology, and its application to the triangulation conjecture.

Finite subgroups of Ham and Symp

Let $(X,ω)$ be a compact symplectic manifold of dimension $2n$ and let $Ham(X,ω)$ be its group of Hamiltonian diffeomorphisms. We prove the existence of a constant $C$, depending on $X$ but not on $ω$, such that any finite subgroup $G\subset Ham(X,ω)$ has an abelian subgroup $A\subseteq G$ satisfying $[G:A]\leq C$, and $A$ can be generated by $n$ elements or fewer. If $b_1(X)=0$ we prove an analogous statement for the entire group of symplectomorphisms of $(X,ω)$. If $b_1(X)\neq 0$ we prove the existence of a constant $C'$ depending only on $X$ such that any finite subgroup $G\subset Symp(X,ω)…

Heegaard diagrams and holomorphic disks

A three-manifold equipped with a Heegaard diagram can be used to set up a Floer homology theory whose differential counts pseudo-holomorphic disks in the $g$-fold symmetric product of the Heegaard surface. This leads to a topological invariant for three-manifolds, Heegaard Floer homology, which is functorial under cobordisms. In this survey article, we sketch this construction and describe some of its topological applications.