G-bordism with singularities andG-homology

Abstract

The bordism and cobordism theories of singular G-manifolds of specified kinds are used to represent various ordinary G-homology and cohomology theories, and their relationship to each other, as well as their relationship to non-singular G-bordism, is studied.1. Introduction. Sullivan once pointed out that ordinary homology may be viewed geometrically as a bordism theory with singularities. This has been formally established by Baas in [1] and by Buoncristiano, Rourke and Sanderson in [3]. Dually, the associated cobordism theories represent ordinary cohomology.Let G be a finite group. One then has several notions of what is meant by ordinary G-cohomology. The first to be proposed was the functor X _» H*(X X G EG) for a G-space X, where EG denotes the universal contractible free G-space. Subsequently, Bredon [2] and Illman [6] described a theory of the following type. Let ^ denote the category whose objects are the G-spaces G/H for subgroups H and whose morphisms G/H -> G/K are the G-equivariant maps. A contravariant coefficient system is then a contravariant functor T from ^to the category of abelian groups. The associated ordinary G-cohomology theory is a generalized G-cohomology theory (see [2]) with dimension axiom of the form

The C2-equivariant cohomology of complex projective spaces

“…Basics. In [8], Hastings and Waner showed that ordinary RO(G)-graded equivariant homology could be defined using bordism of manifolds with a certain kind of singularity, similar to the Baas-Sullivan approach nonequivariantly. The Hastings-Waner result can be generalized to show that ordinary RO(ΠB)-graded homology can be given by bordism of singular manifolds as well.…”