Construction of a linear K-system in Hamiltonian Floer theory

Abstract

© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.The notion of linear K-system was 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 (Fukaya et al. in Springer monographs in mathematics, Springer, Berlin, 2020). 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 Fukaya et al. (Springer monographs in mathematics, Springer, Berlin, 2020)] are the construction of Floer cohomology for periodic Hamiltonian systems on general compact symplectic manifolds without any restriction, and the 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 pseudoholomorphic curves established in our earlier papers and books. However, the paper itself is geometric in nature, and does not presume much prior knowledge of Kuranishi structures and their construction but assumes only the elementary part thereof, and results from Fukaya et al. (Surv Differ Geom 22:133–190, 2018) and Fukaya et al. (Exponential decay estimate and smoothness of the moduli space of pseudoholomorphic curves) on their 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.