$RO(\Pi B)$-Graded Ordinary Homology and Cohomology

Abstract

As we’ve already mentioned, R O(G)-graded ordinary homology is not adequate to give Poincaré duality for G-manifolds except in the case of manifolds modeled on a single representation. To get Poincaré duality for general G-manifolds, we need to extend to a theory indexed on representations of $\Pi X$. (For simplicity of notation we shall now write $\Pi X$ for $\Pi _{G}X$.) That is, the homology and cohomology of X should be graded on representations of $\Pi X$. A construction of the $RO(\Pi X)$-graded theory for finite G was given in Costenoble and Waner (Michigan Math J 39:325–351, 1992), and the theory was used in Costenoble and Waner (Michigan Math J 39:415–424, 1992) and Michigan Math J (40:577–604, 1993) to obtain π–π theorems for equivariant Poincaré duality spaces and equivariant simple Poincaré duality spaces. In this chapter we give the construction for all compact Lie groups.