Construction of a linear K-system in Hamiltonian Floer theory

TL;DR

This paper constructs a geometric model called a linear K-system for Hamiltonian Floer theory, enabling the definition of Floer cohomology on general symplectic manifolds and establishing an isomorphism with singular cohomology.

Contribution

It provides a geometric realization of the linear K-system in the Morse-Bott setting, facilitating Floer cohomology construction without restrictions.

Findings

Construction of Floer cohomology for general compact symplectic manifolds.

Establishment of an isomorphism between Floer cohomology and singular cohomology.

Development of a detailed inductive procedure for constructing linear K-systems.

Abstract

The notion of linear K-system is introduced by the present authors as an abstract model arising from the structure of compactified moduli spaces of solutions to Floer's equation in the book [FOOO14]. The purpose of the present article is to provide a geometric realization of the linear K-system associated with solutions to Floer's equation in the Morse-Bott setting. Immediate consequences (when combined with the abstract theory from [FOOO14]) are construction of Floer cohomology for periodic Hamiltonian system on general compact symplectic manifold without any restriction, and construction of an isomorphism over the Novikov ring between the Floer cohomology and the singular cohomology of the underlying symplectic manifold. The present article utilizes various analytical results on pseudo-holomorphic curves established in our earlier papers and books. However the paper itself is geometric in nature, and does not presume the readers' prior knowledge much on Kuranishi structure and its construction but assumes only the elementary part of thereof, and results from [FOOO11] and [FOOO,Chapter 8] on its construction, and the standard knowledge on Hamiltonian Floer theory. We explain the general procedure of the construction of a linear K-system by explaining in detail the inductive steps of ensuring the compatibility conditions for the system of Kuranishi structures leading to a linear K-system for the case of Hamiltonian Floer theory. A certain minor errors are corrected in this version. (to appear in the special volume dedicated to Claude Viterbo of Journal of Fixed Point Theory and Applications.)