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

Abstract

In this paper we describe and continue the study begun in [5] 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 in [5]. 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.

Floer homotopy theory, revisited

“…One does start with geometric information that allows for the definition of a chain complex, but knowing if this complex comes, in a natural way from the data of the Floer theory, is not at all clear, and was the central question of study in [12]. This homotopy theoretic question was addressed more specifically in [10]. Of central importance in this study was to understand how the attaching maps of the cells in a finite CW -spectrum can be understood geometrically, via the theory of (framed) cobordism of manifolds with corners.…”

“…Clearly |Z| will have one cell of dimension i for every element of π 0 (Z(i)) = π 0 (E i ) = B i . The attaching maps were described in [12], [9], [10] in the following way.…”

“…In the case when the CW -structure comes from a Morse function f : M n → R on a closed n-dimensional Riemannian manifold, the attaching maps φ p,q defining a functor Z f : J 0 → Spectra, were shown in [9], [10] to come from the cobordism-type of the moduli spaces of gradient flow lines connecting critical points of index p to those of index q. The relevant cobordism theory is framed cobordism of manifolds with corners.…”

“…Thus to extend this idea to the Floer setting, with the goal of developing a "Floer homotopy type", one needs to understand certain cobordism-theoretic properties of the corresponding moduli spaces of gradient flows of the particular Floer theory. (In [10] the author also considered how other cobordism theories give rise not to "Floer homotopy types" but rather to "Floer module spectra", or said another way, "Floer E-theory" where E • is a generalized homology theory. We refer the reader to [10] for details.)…”

“…(In [10] the author also considered how other cobordism theories give rise not to "Floer homotopy types" but rather to "Floer module spectra", or said another way, "Floer E-theory" where E • is a generalized homology theory. We refer the reader to [10] for details.) In order to make this idea precise we recall some basic facts about cobordisms of manifolds with corners.…”