Positive Scalar Curvature and Poincare Duality for Proper Actions

TL;DR

This paper investigates G-equivariant index theory for proper co-compact actions of almost-connected Lie groups, exploring obstructions to positive scalar curvature, rigidity results, and Poincare duality in equivariant K-theory.

Contribution

It extends index theory and Poincare duality results to proper actions of almost-connected Lie groups, including new obstructions and rigidity theorems.

Findings

Established Poincare duality for G-equivariant K-homology and K-theory in specific cases.

Proved a rigidity result for almost-complex manifolds generalising Hattori's work.

Identified obstructions and conditions for G-invariant positive scalar curvature metrics.

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 Poincare duality between G-equivariant K-homology and K-theory, observing that Poincare duality does not necessarily hold for general G.