Local index theory and ℤ/kℤ K-theory

2024/10/21 by Man-Ho Ho, Ho, Man-Ho · 1 voice
Mathematics · #19K56 #19L10 #19L50 #58J20 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #K-Theory and Homology (math.KT) #math.DG #math.KT

paper · pdf · doi:10.48550/arxiv.2410.16399

openalex publication_date 2024/10/21 · arxiv published 2024/10/21 · arxiv updated 2025/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For any given submersion π:X→ B with closed, oriented and spin^c fibers of even dimension, equipped with a Riemannian and differential spin^c structure, we apply the Atiyah-Singer-Gorokhovsky-Lott approach to the local family index theorem without the kernel bundle assumption to construct an analytic index \textrmind^a_k in odd ℤ/kℤ K-theory at the cocycle level. This is achieved by associating to every cocycle (E, F, α) of the odd ℤ/kℤ K-theory group of X a cocycle \textrmind^a_k(E, F, α) of the odd ℤ/kℤ K-theory group of B. We also prove a Riemann-Roch-Grothendieck-type formula in odd ℤ/kℤ K-theory, which expresses the Cheeger-Chern-Simons form of \textrmind^a_k(E, F, α) in terms of that of (E, F, α). Furthermore, we show that the analytic index \textrmind^a_k and the Riemann-Roch-Grothendieck-type formula in odd ℤ/kℤ K-theory refine the underlying geometric bundle of the analytic index and the Riemann-Roch-Grothendieck theorem in ℝ/ℤ K-theory, respectively.

Discussions

• Man-Ho Ho Local index theory and $\mathbb{Z}/k\mathbb{Z}$ $K$-theory https://arxiv.org/abs/2410.16399 [bsky, 0 points, 0 comments]

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