An equivariant Poincar\'e duality for proper cocompact actions by matrix groups

TL;DR

This paper establishes a Poincaré duality in equivariant K-theory and K-homology for proper cocompact actions of matrix groups on G-spin^c manifolds, extending duality concepts in equivariant topology.

Contribution

It proves an equivariant Poincaré duality for proper cocompact actions by matrix groups, connecting G-equivariant K-theory and K-homology via geometric models.

Findings

Poincaré duality holds between G-equivariant K-theory and K-homology.

Duality is established for actions with compact quotient.

The result applies to G-spin^c manifolds with proper isometric actions.

Abstract

Let $G$ be a linear Lie group acting properly and isometrically on a $G$-spin$^c$ manifold $M$ with compact quotient. We show that Poincar\'e 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.