Positive scalar curvature and Poincaré duality for proper actions

Abstract

For $G$ an almost-connected Lie group, we study $G$-equivariant index theory for proper co-compact actions with various applications, including obstructions to and existence of $G$-invariant Riemannian metrics of positive scalar curvature. We prove a rigidity result for almost-complex manifolds, generalising Hattori’s results, and an analogue of Petrie’s conjecture. When $G$ is an almost-connected Lie group or a discrete group, we establish Poincaré duality between $G$-equivariant $K$-homology and $K$-theory, observing that Poincaré duality does not necessarily hold for general $G$.