Floer homotopy theory, realizing chain complexes by module spectra, and manifolds with corners

TL;DR

This paper extends Floer homotopy theory by showing how smooth, oriented moduli spaces with corners can realize Floer complexes as module spectra, enabling Floer homology theories over generalized cohomology.

Contribution

It generalizes the realization of Floer complexes as module spectra to cases with manifolds with corners and orientations in generalized cohomology, broadening the scope of Floer homotopy theory.

Findings

Floer complexes can be realized as E-module spectra when moduli spaces are smooth and oriented.

A functorial framework for realizing chain complexes by module spectra is developed.

Manifolds with corners are suitable for defining orientations in this context.

Abstract

In this paper we describe and continue the study begun by the author, Jones, and Segal, of the homotopy theory that underlies Floer theory. In that paper the authors addressed the question of realizing a Floer complex as the celluar chain complex of a CW -spectrum or pro-spectrum, where the attaching maps are determined by the compactified moduli spaces of connecting orbits. The basic obstructions to the existence of this realization are the smoothness of these moduli spaces, and the existence of compatible collections of framings of their stable tangent bundles. In this note we describe a generalization of this, to show that when these moduli spaces are smooth, and are oriented with respect to a generalized cohomology theory E^*, then a Floer E_* -homology theory can be defined. In doing this we describe a functorial viewpoint on how chain complexes can be realized by E -module spectra, generalizing the stable homotopy realization criteria given earlier by the author, Jones, and Segal. Since these moduli spaces, if smooth, will be manifolds with corners, we give a discussion about the appropriate notion of orientations of manifolds with corners.