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Index invariants and Eta invariants determine Differential KO theory in degrees that are multiples of 8

Tan Su

TL;DR

The paper develops a degree $0mod 8$ differential KO-theory through $R/ Z$-valued eta invariants and $KO$-index data, culminating in a differential KO-character that determines real vector bundles with orthogonal connections up to Chern–Simons equivalence. It builds a robust hexagon relating the structured bundle model to Freed–Lott style differential KO, and proves an isomorphism between the differential KO theory and differential KO characters, via a uniqueness philosophy. Two family index theorems are established for Riemannian submersions with $8k$-dimensional spin fibers, showing that pushforwards in the structured-bundle and KO-character models agree, with the Cheeger–Bismut adiabatic limit explaining their compatibility. The work also connects to classical invariants such as Adams's $e$-invariant and analyzes invariants for flat bundles, thereby providing a comprehensive framework for differential KO theory in degrees multiple of eight and its geometric and topological consequences.

Abstract

Sullivan--Simons developed a Cheeger--Simons differential character analogue for degree (0 mod 2) differential K-theory, giving a complete set of numerical invariants that determine a complex vector bundle with unitary connection on a base manifold X, up to Chern--Simons equivalence of the connection. In this paper we develop a degree (0 mod 8) differential KO-analogue. Namely, given a real vector bundle with orthogonal connection, we construct R/Z -valued eta-invariants in the context of Atiyah--Patodi--Singer and Z2 Atiyah--Singer index invariants that completely determine differential KO-theory in degree (0 mod 8); we call this the differential KO-character. In the second part, for a Riemannian submersion X to B with closed 8k-dimensional spin fibers, we develop two family index theorems in differential KO -theory: one in the differential KO -character model and one in the structured-bundle model. The fact that these two pushforwards agree follows from the Bismut--Cheeger adiabatic limit theorem, providing a new interpretation of that result.

Index invariants and Eta invariants determine Differential KO theory in degrees that are multiples of 8

TL;DR

The paper develops a degree differential KO-theory through -valued eta invariants and -index data, culminating in a differential KO-character that determines real vector bundles with orthogonal connections up to Chern–Simons equivalence. It builds a robust hexagon relating the structured bundle model to Freed–Lott style differential KO, and proves an isomorphism between the differential KO theory and differential KO characters, via a uniqueness philosophy. Two family index theorems are established for Riemannian submersions with -dimensional spin fibers, showing that pushforwards in the structured-bundle and KO-character models agree, with the Cheeger–Bismut adiabatic limit explaining their compatibility. The work also connects to classical invariants such as Adams's -invariant and analyzes invariants for flat bundles, thereby providing a comprehensive framework for differential KO theory in degrees multiple of eight and its geometric and topological consequences.

Abstract

Sullivan--Simons developed a Cheeger--Simons differential character analogue for degree (0 mod 2) differential K-theory, giving a complete set of numerical invariants that determine a complex vector bundle with unitary connection on a base manifold X, up to Chern--Simons equivalence of the connection. In this paper we develop a degree (0 mod 8) differential KO-analogue. Namely, given a real vector bundle with orthogonal connection, we construct R/Z -valued eta-invariants in the context of Atiyah--Patodi--Singer and Z2 Atiyah--Singer index invariants that completely determine differential KO-theory in degree (0 mod 8); we call this the differential KO-character. In the second part, for a Riemannian submersion X to B with closed 8k-dimensional spin fibers, we develop two family index theorems in differential KO -theory: one in the differential KO -character model and one in the structured-bundle model. The fact that these two pushforwards agree follows from the Bismut--Cheeger adiabatic limit theorem, providing a new interpretation of that result.

Paper Structure

This paper contains 46 sections, 2 theorems, 201 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Structured Bundle in Differential KO theory
  3. Proposition 2.2
  4. Definition 2.3
  5. Definition 2.4
  6. Definition 2.5
  7. Theorem 2.6
  8. Definition 2.7
  9. Definition 2.8
  10. Proposition 2.9
  11. Proposition 2.10
  12. Theorem 2.11
  13. Freed-Lott's Differential KO
  14. Definition 3.1
  15. Differential KO character
  16. ...and 31 more sections

Key Result

Theorem 8.1

The $e$-invariant of a framed $4k-1$-dimensional manifold $(M,\pi)$ is given by where $\varepsilon(k)=1$ if $k$ is even and $\varepsilon(k)=\tfrac{1}{2}$ if $k$ is odd.

Figures (1)

  • Figure 1: Example of a $\mathbb{Z}_{3}$-manifold.

Theorems & Definitions (22)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof : Proof of Theorem 4.4
  • proof
  • proof
  • proof
  • proof
  • ...and 12 more