  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.
