Equivariant Spectral Flow for Families of Dirac-type Operators

TL;DR

This paper develops an equivariant theory of spectral flow for families of Dirac-type operators under a proper cocompact action of a unimodular group $G$, with the flow valued in the $K$-theory group $K_0(C^*(G))$. It builds a $K$K-theoretic framework, proving that the equivariant spectral flow refines the classical spectral flow via an integration map and satisfies index theorems equating spectral flow with the Fredholm index of a Dirac–Schrödinger operator. It then connects the flow to higher secondary invariants, establishing relations with delocalised $\,η$-invariants and $\,ρ$-invariants for varying positive scalar curvature metrics, and derives higher APS-type index formulae in this equivariant setting. These results extend index theory on noncompact spaces with group actions, enabling new invariants and comparison results for metrics and topology in geometric analysis and noncommutative geometry.

Abstract

In the setting of a proper, cocompact action by a locally compact, unimodular group $G$ on a Riemannian manifold, we construct equivariant spectral flow of paths of Dirac-type operators. This takes values in the $K$-theory of the group $C^*$-algebra of $G$. In the case where $G$ is the fundamental group of a compact manifold, the summation map maps equivariant spectral flow on the universal cover to classical spectral flow on the base manifold. We obtain "index equals spectral flow" results. In the setting of a smooth path of $G$-invariant Riemannian metrics on a $G$-spin manifold, we show that the equivariant spectral flow of the corresponding path of spin Dirac operators relates delocalised $η$-invariants and $ρ$-invariants for different positive scalar curvature metrics to each other.

Theorems & Definitions (108)

• Definition 2.1
• Definition 2.2: wahl
• Definition 2.3
• Definition 2.4
• Definition 2.5: Roe96; WillettYu; piazzaschick
• Definition 2.6
• Remark 2.7
• Remark 3.1
• Remark 3.2
• Lemma 3.3