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同源性理论。我们感兴趣的是它的版本的自由回路空间的基础辛流形所产生的不动点的哈密顿同态。最初，为了证明Arnold猜想，人们在闭辛流形上发展了Floer同调，利用它总是同构于普通同调。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子环结构。拟议项目由不同部分组成。一个是有关的S1-等变Floer同调理论的行动reparametrizing循环。此外，环形结构仍有待建立和分析。计划寻找与自由回路空间的拓扑结构的更本质的关系。这需要考虑类的非紧辛流形，也将考虑局部版本的弗洛尔同调连同其环结构。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/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.