The $S^1$-Equivariant signature for semi-free actions as an index formula

TL;DR

This paper constructs a Dirac-type operator on the orbit space of a semi-free $S^1$-action that computes the $S^1$-equivariant signature, extending previous work by John Lott and addressing both Witt and non-Witt cases.

Contribution

It introduces a new Dirac-type operator on the orbit space that matches Lott's $S^1$-equivariant signature and explores its properties in Witt and non-Witt stratified spaces.

Findings

The operator's index equals Lott's signature under Witt conditions.

The operator remains essentially self-adjoint in non-Witt cases.

Conjecture that the operator's index computes the $S^1$-signature in general.

Abstract

John Lott defined an integer-valued signature $\sigma_{S^1}(M)$ for the orbit space of a compact orientable manifold with a semi-free $S^1$-action but he did not construct a Dirac-type operator which has this signature as its index. We construct such operator on the orbit space and we show that it is essentially unique and that its index coincides with Lott's signature, at least when the stratified space satisfies the so-called Witt condition. For the non-Witt case, this operator remains essentially self-adjoint (in contrast to the Hodge de-Rham operator) and it has a well defined index which we conjecture will also compute $\sigma_{S^1}(M)$. This article is a condensed version of the original author's PhD Thesis where the theory of induced Dirac-Schr\"odinger-type operators is developed in detail.