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

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 $0$.</jats:p

On the Product in Negative Tate Cohomology for Finite Groups

Our aim in this paper is to give a geometric description of the cup product in negative degrees of Tate cohomology of a finite group with integral coefficients. By duality it corresponds to a product in the integral homology of $BG$: {$$H_{n}(BG,\mathbb{Z})\otimes H_{m}(BG,\mathbb{Z})\rightarrow H_{n+m+1}(BG,\mathbb{Z})$$} for $n,m>0$. We describe this product as join of cycles, which explains the shift in dimensions. Our motivation came from the product defined by Kreck using stratifold homology. We then prove that for finite groups the cup product in negative Tate cohomology and the Kreck product coincide. The Kreck product also applies to the case where $G$ is a compact Lie group (with an additional dimension shift).Comment: 13 page

Me, Myself and I: Aggregated and Disaggregated Identities on Social Networking Services

In this article I explore some of the legal issues arising from the transformation of SNS operators to providers of digital identity. I consider the implications of the involvement of private sector entities in the field of identity management and discuss some of the privacy implications, as well as the prospects for conciliation between online anonymity and pseudonymity, on the one hand, and the need for identifiability and accountability on the other hand.

A geometric description of the Atiyah–Hirzebruch spectral sequence for B-bordism

In this paper we give a geometric description of the general term and the differential of the Atiyah–Hirzebruch spectral sequence for B-bordism. This description is given in terms of bordism classes of maps from stratifolds. We illustrate that with a computational example. We also discuss the case of a general homology theory, where this description is given in terms of the Postnikov sections of the given theory