Semi-infinite Homology of Floer spaces

Abstract

This dissertation presents a framework for defining Floer homology of infinite-dimensional spaces with a functional. This approach is meant to generalize the traditional constructions of Floer homologies which mimic the construction of the Morse-Smale-Witten complex. To define Floer homology we use cycles modelled on infinitedimensional manifolds with corners, as described by Maksim Lipyanskiy, where the key is to impose appropriate compactness and polarization axioms on the cycles. We separate and carefully inspect these two types of axioms, paying special attention to correspondences, generalizing the definition of a polarization as well as axiomatizing the notion of a family of perturbations. The latter is used to define an intersection pairing and maps induced on Floer homology by correspondences. Moreover, we prove suspension isomorphisms and prove that this Floer homology agrees with Morse homology for finite-dimensional manifolds with a Palais-Smale functional. Finally, we explain how to apply this framework to Seiberg-Witten-Floer theory, defining Floer homology groups associated to rational homology spheres and their spinc-structures. Importantly, we prove moduli spaces of solutions to Seiberg-Witten equations induce maps on Floer homology in a functorial fashion.