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In this paper we use tools from differential topology to give a geometricdescription of cohomology for Hilbert manifolds. Our model is Quillen'sgeometric description of cobordism groups for finite dimensional smoothmanifolds \cite{Q}. Quillen stresses the fact that this construction allows thedefinition of Gysin maps for "oriented" proper maps. For finite dimensionalmanifolds one has a Gysin map in singular cohomology which is based onPoincar\'e duality, hence it is not clear how to extend it to infinitedimensional manifolds. But perhaps one can overcome this difficulty by giving aQuillen type description of singular cohomology for Hilbert manifolds. This iswhat we do in this paper. Besides constructing a general Gysin map, one of ourmotivations was a geometric construction of equivariant cohomology, which evenfor a point is the cohomology of the infinite dimensional space $BG$, which hasa Hilbert manifold model. Besides that, we demonstrate the use of such ageometric description of cohomology by several other applications. We give aquick description of characteristic classes of a finite dimensional vectorbundle and apply it to a generalized Steenrod representation problem forHilbert manifolds and define a notion of a degree of proper oriented Fredholmmaps of index $0$.

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