Topological categories related to Fredholm operators: II. The analytical index

TL;DR

This paper introduces a new, more flexible definition of the analytic index for families of Fredholm operators, extending classical concepts and clarifying spectral section subtleties, with proofs of equivalence to Atiyah-Singer under certain conditions.

Contribution

It develops a weaker continuity-based definition of the analytic index for Fredholm families, compatible with Hilbert bundles, and proves its consistency with the classical Atiyah-Singer index.

Findings

New analytic index definition under weaker assumptions

Equivalence with Atiyah-Singer index when applicable

Clarification of spectral sections by Melrose and Piazza

Abstract

Naively, the analytic index of a family of self-adjoint Fredholm operators ought to be (an equivalence class of) the family of the kernels of these operators. The present paper is devoted to a rigorous version of this idea based on ideas of Segal as developed by the author in arXiv:2111.14313 [math.KT]. The resulting new definition of the analytic index makes sense under much weaker continuity assumptions than the Atiyah-Singer one and can be easily adjusted to families of operators in fibers of a Hilbert bundle. We prove the correctness of the new definition and show that it agrees with the Atiyah-Singer one when the latter applies. As an illustration, these results are used to clarify some subtle aspects of the notion of spectral sections introduced by Melrose and Piazza. The necessary definitions and results from arXiv:2111.14313 [math.KT] are repeated or reviewed in order to make this paper independent to the extent possible.