Equivariant ordinary homology and cohomology

Details Equivariant ordinary homology and cohomology Equivariant ordinary homology and cohomology

2003-10-16

arxiv.org/abs/math/0310237v2

Mathematics

Algebraic Topology

141 pages, 1 figure, AMSLaTeX; substantional revision of the original

Scientific paper

Poincare duality lies at the heart of the homological theory of manifolds. In the presence of the action of a group it is well-known that Poincare duality fails in Bredon's ordinary, integer-graded equivariant homology. We give here a detailed account of one way around this problem, which is to extend equivariant ordinary homology to a theory graded on representations of fundamental groupoids. Versions of this theory have appeared previously for actions of finite groups, but this is the first account that works for all compact Lie groups. The first part of this work is a detailed discussion of RO(G)-graded ordinary homology and cohomology, collecting scattered results and filling in gaps in the literature. The remainder of the work discusses the extension to grading on representations of fundamental groupoids, concentrating on those aspects that are not simple generalizations of the RO(G)-graded case. These theories can be viewed as defined on parametrized spaces, and then the representing objects are parametrized spectra; we use heavily recent foundational work of May and Sigurdsson on parametrized spectra. While it is well-known that parametrized cohomology theories are represented by parametrized spectra, we give what we believe to be the first discussion of how parametrized homology theories are represented by such spectra. We end with a discussion of Poincare duality for arbitrary smooth equivariant manifolds.

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