An equivariant Poincaré duality for proper cocompact actions by matrix groups

Abstract

Let $G$ be a linear Lie group acting properly on a $G$-${\mathrm{spin}}^c$ manifold $M$ with compact quotient. We give a short proof that Poincaré duality holds between $G$-equivariant $K$-theory of $M$, defined using finite-dimensional $G$-vector bundles, and $G$-equivariant $K$-homology of $M$, defined through the geometric model of Baum and Douglas.