Topological categories related to Fredholm operators: II. The analytic index
Abstract
Naively, the analytic index of a family of selfadjoint 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 AtiyahSinger 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 AtiyahSinger 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2021
 arXiv:
 arXiv:2111.15081
 Bibcode:
 2021arXiv211115081I
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Algebraic Topology;
 Mathematics  KTheory and Homology;
 58J20;
 19K56 (Primary);
 46L80;
 47A53;
 47F10 (Secondary)
 EPrint:
 49 pages. The statement and the proof of Theorem 7.2 are corrected. This caused minor changes in Section 8