  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. In particular, we give details on change of groups and products
that do not seem to have appeared elsewhere. We also discuss the relationship
between ordinary homology and cohomology when the group is compact Lie, in
which case the two theories are not represented by the same spectrum. 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 foundational work of May and Sigurdsson on
parametrized spectra. We end with a discussion of Poincare duality for
arbitrary smooth equivariant manifolds.
