C*-Algebraic Higher Signatures and an Invariance Theorem in Codimension Two

TL;DR

This paper develops a new approach to signature classes in C*-algebra K-theory, proving invariance under certain homotopy equivalences for noncompact manifolds and establishing a codimension two vanishing theorem for Dirac operator indices.

Contribution

It introduces a variation in the construction of signature classes in C*-algebra K-theory, enabling proofs of invariance under specific homotopy equivalences for noncompact manifolds.

Findings

Proved equality of signature classes under certain homotopy conditions.

Established a codimension two vanishing theorem for Dirac operator indices.

Extended Gromov-Lawson's work to noncompact manifolds with new invariance results.

Abstract

We revisit the construction of signature classes in C*-algebra K-theory, and develop a variation that allows us to prove equality of signature classes in some situations involving homotopy equivalences of noncompact manifolds that are only defined outside of a compact set. As an application, we prove a counterpart for signature classes of a codimension two vanishing theorem for the index of the Dirac operator on spin manifolds (the latter is due to Hanke, Pape and Schick, and is a development of well-known work of Gromov and Lawson).