Analysis of Floer Homology, its Natural Ring Structure and its S1-equivariant version, in Relation with the Free Loop Space and Symplectic Invariants

该项目的目标是Floer同调理论。我们感兴趣的是由哈密顿微分同胚的不动点生成的基础辛流形的自由环空间的形式。Floer同调最初是为了证明Arnold猜想而发展起来的，它利用了Floer同调总是与普通同调同构的性质。随后的进一步发展也显示了编码在其中的基本辛不变量。特别令人感兴趣的是自然的裤环结构。拟议的项目由不同的部分组成。一个是关于环的重参数化作用的S1-等变Floer同调理论。此外，环结构还有待建立和分析。计划找到与自由环空间的拓扑结构更本质的联系。这需要考虑非紧辛流形的类，并且还将考虑Floer同调的局部化形式及其环结构。此外，该项目还要求S1-等变Floer同调和接触同调之间的精确关系，特别是对于具有各自辛结构/接触结构的余切/单位余切丛的情况。

The target of the proposed project is the theory of Floer homology. We are interested in its version for the free loop space of the underlying symplectic manifold generated by fixed points of Hamiltonian diffeomorphisms. Originally, Floer homology had been developed for closed symplectic manifold in order to prove Arnold's conjecture, exploiting that it is always isomorphic to the ordinary homology. Subsequent further developments showed also essential symplectic invariants encoded in it. Of particular interest is the natural pair-of-pants ring structure. The proposed project consists of different parts. One is concerned with the S1-equivariant Floer homology theory with respect to the action by reparametrizing loops. There also, the ring structure has still to be established and analyzed. It is planned to find more essential relations with the topological structure of the free loop space. This requires to consider classes of non-compact symplectic manifolds, and also a localized version of Floer homology together with its ring structure will be considered. Moreover, the project also asks for a precise relation between S1-equivariant Floer homology and contact homology, in particular for the case of cotangent/unit cotangent bundles with the respective symplectic/contact structure.