The Odd Dimensional Analogue of a Theorem of Getzler and Wu

TL;DR

This paper extends the higher Atiyah-Patodi-Singer index theorem to odd-dimensional manifolds with boundary within the $b$-calculus framework, introducing a counterpart to the eta invariant for even-dimensional closed manifolds.

Contribution

It provides the first odd-dimensional analogue of Getzler and Wu's theorem, broadening the understanding of index theory in the $b$-calculus setting.

Findings

Established an index theorem for odd-dimensional manifolds with boundary

Defined a natural eta invariant counterpart for even-dimensional closed manifolds

Extended the applicability of the higher Atiyah-Patodi-Singer index theorem

Abstract

We prove an analogue for odd dimensional manifolds with boundary, in the $b$-calculus setting, of the higher Atiyah-Patodi-Singer index theorem by Getzler and Wu, thus obtain a natural counterpart of the eta invariant for even dimensional closed manifolds.