Higher genera for proper actions of Lie groups

TL;DR

This paper develops index formulas for higher indices of Dirac operators on G-proper manifolds with applications to higher genera, homotopy invariance, and scalar curvature obstructions in the context of Lie groups with specific geometric properties.

Contribution

It introduces new index formulas for G-equivariant Dirac operators on manifolds with proper G-actions, linking geometric invariants to group properties.

Findings

Higher G-homotopy invariance of higher signatures.

Vanishing of A-hat genera under positive scalar curvature.

Index formulas applicable to Lie groups with RD property and non-positive curvature.

Abstract

Let G be a Lie group with finitely many connected components and let K be a maximal compact subgroup. We assume that G satisfies the rapid decay (RD) property and that G/K has non-positive sectional curvature. As an example, we can take G to be a connected semisimple Lie group. Let M be a G-proper manifold with compact quotient M/G. In this paper we establish index formulae for the C^*-higher indices of a G-equivariant Dirac-type operator on M. We use these formulae to investigate geometric properties of suitably defined higher genera on M. In particular, we establish the G-homotopy invariance of the higher signatures of a G-proper manifold and the vanishing of the A-hat genera of a G-spin, G-proper manifold admitting a G-invariant metric of positive scalar curvature.