Floer Homotopy Theory, Realizing Chain Complexes by Module Spectra, and Manifolds with Corners

In this paper we describe and continue the study begun in Cohen et al. (Progress in Mathematics, vol. 133, Birkhauser, Boston, 1995, pp. 287-325) of the homotopy theory that underlies Floer theory. In that paper the authors addressed the question of realizing a Floer complex as the cellular 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 in Cohen et al. (Progress in Mathematics, vol. 133, Birkhauser, Boston, 1995, pp. 287-325). 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.