Spectral flow, index and the signature operator

TL;DR

This paper explores the relationship between spectral flow and index theory for operators in von Neumann algebras, proving invariance of the tangential signature under metric changes on foliated manifolds.

Contribution

It generalizes spectral flow-index relations to semifinite von Neumann algebras and proves metric invariance of the tangential signature for foliated manifolds.

Findings

Spectral flow relates to the index for paths of selfadjoint operators in von Neumann algebras.

The von Neumann spectral flow vanishes for the tangential signature operator under metric variation.

The tangential signature of a foliated manifold with boundary is independent of the metric.

Abstract

We relate the spectral flow to the index for paths of selfadjoint Breuer-Fredholm operators affiliated to a semifinite von Neumann algebra, generalizing results of Robbin-Salamon and Pushnitski. Then we prove the vanishing of the von Neumann spectral flow for the tangential signature operator of a foliated manifold when the metric is varied. We conclude that the tangential signature of a foliated manifold with boundary does not depend on the metric. In the Appendix we reconsider integral formulas for the spectral flow of paths of bounded operators.