Index theory in spaces of manifolds

TL;DR

This paper generalizes the Atiyah-Singer family index theorem within the framework of spaces of manifolds, extending to Dirac-type operators over arbitrary C*-algebras, advancing the mathematical understanding of index theory.

Contribution

It introduces a broad generalization of the Atiyah-Singer index theorem in the setting of spaces of manifolds, applicable to Dirac operators over any C*-algebra.

Findings

Generalized index theorem for spaces of manifolds

Applicable to Dirac operators over arbitrary C*-algebras

Extends the mathematical framework of index theory

Abstract

We formulate and prove a generalization of the Atiyah-Singer family index theorem in the context of the theory of spaces of manifolds \`a la Madsen, Tillmann, Weiss, Galatius and Randal-Williams. Our results are for Dirac-type operators linear over arbitrary $C^*$-algebras.