HILBERT STRATIFOLDS AND A QUILLEN TYPE GEOMETRIC DESCRIPTION OF COHOMOLOGY FOR HILBERT MANIFOLDS. (27th February 2018)

• Record Type: Journal Article
• Authors: KRECK, MATTHIAS; TENE, HAGGAI
• Is Part Of: Forum of mathematics. Volume 6(2018)
• Publication Date: 2018-02-27
• Subjects: 58B05 (primary), -- 57R19 (secondary); Mathematics -- Periodicals; 510
• DOI: 10.1017/fms.2018.1
• Languages: English
• ISSNs: 2050-5094

Abstract

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 [Quillen, 'Elementary proofs of some results of cobordism theory using steenrod operations', Adv. Math., 7 (1971)]. 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 Poincaré 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 $0$ .