Product formula for Atiyah-Patodi-Singer index classes and higher signatures

TL;DR

This paper develops a product formula for Atiyah-Patodi-Singer index classes and extends it to higher signatures, providing new tools for analyzing Dirac operators on manifolds with boundary in the context of C*-algebras.

Contribution

It introduces generalized product-type boundary conditions for Dirac operators and proves a K-theoretic product formula, extending the topological signature to higher signatures.

Findings

Established a product formula for K-theoretic index classes.

Generalized the product formula for higher signatures.

Applied the formula to Dirac operators on product manifolds.

Abstract

We define generalized Atiyah-Patodi-Singer boundary conditions of product type for Dirac operators associated to C*-vector bundles on the product of a compact manifold with boundary and a closed manifold. We prove a product formula for the K-theoretic index classes, which we use to generalize the product formula for the topological signature to higher signatures.