Hilbert stratifolds and a Quillen type geometric description of cohomology for Hilbert manifolds

TL;DR

This paper develops a geometric framework for cohomology of Hilbert manifolds inspired by Quillen's cobordism model, enabling Gysin maps and applications like equivariant cohomology and characteristic classes in infinite dimensions.

Contribution

It introduces a Quillen-type geometric description of cohomology for Hilbert manifolds, extending finite-dimensional tools to infinite-dimensional settings.

Findings

Constructed a Gysin map for Hilbert manifolds.

Provided a geometric approach to equivariant cohomology.

Applied the framework to characteristic classes and Fredholm maps.

Abstract

In this paper we use tools from differential topology to give a geometric description of cohomology for Hilbert manifolds. Our model is Quillen's geometric description of cobordism groups for finite dimensional smooth manifolds \cite{Q}. Quillen stresses the fact that this construction allows the definition of Gysin maps for "oriented" proper maps. For finite dimensional manifolds one has a Gysin map in singular cohomology which is based on Poincar\'e duality, hence it is not clear how to extend it to infinite dimensional manifolds. But perhaps one can overcome this difficulty by giving a Quillen type description of singular cohomology for Hilbert manifolds. This is what we do in this paper. Besides constructing a general Gysin map, one of our motivations was a geometric construction of equivariant cohomology, which even for a point is the cohomology of the infinite dimensional space $BG$, which has a Hilbert manifold model. Besides that, we demonstrate the use of such a geometric description of cohomology by several other applications. We give a quick description of characteristic classes of a finite dimensional vector bundle and apply it to a generalized Steenrod representation problem for Hilbert manifolds and define a notion of a degree of proper oriented Fredholm maps of index $0$.