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A Product Formula for Family Indices and Family Band Width Estimates

Chenkai Song

TL;DR

The paper extends Gromov's sharp width bound for bands of positive scalar curvature to the setting of fiber bundles with infinite family Â-area by developing a family index theory framework for twisted Dirac operators. A central contribution is a family product formula, proven separately for even and odd fiber dimensions, which identifies the index bundle of Callias-type deformations with the index bundle of the underlying twisted Dirac operator family in K-theory. Using this product formula, the authors prove a sharp, uniform band width estimate in the family context, employing distance functions, Callias-type operators, Friedrich inequalities, and an ODE-based auxiliary construction. The results connect geometric width bounds to family index theory and broaden the applicability of Dirac-operator techniques in geometric analysis of fiber bundles. The framework also clarifies when the infinite Â-area condition guarantees nonvanishing family indices, enabling the band width results to extend beyond previously known cases.

Abstract

We extend Gromov's conjecture on the sharp width estimate for Riemannian bands with positive scalar curvature to the family case and prove that it holds for fiber bundles with infinite family A-hat area. The method we employ is based on Dirac operators and the family index theory. Our proof relies on a product formula for index bundles established in this paper.

A Product Formula for Family Indices and Family Band Width Estimates

TL;DR

The paper extends Gromov's sharp width bound for bands of positive scalar curvature to the setting of fiber bundles with infinite family Â-area by developing a family index theory framework for twisted Dirac operators. A central contribution is a family product formula, proven separately for even and odd fiber dimensions, which identifies the index bundle of Callias-type deformations with the index bundle of the underlying twisted Dirac operator family in K-theory. Using this product formula, the authors prove a sharp, uniform band width estimate in the family context, employing distance functions, Callias-type operators, Friedrich inequalities, and an ODE-based auxiliary construction. The results connect geometric width bounds to family index theory and broaden the applicability of Dirac-operator techniques in geometric analysis of fiber bundles. The framework also clarifies when the infinite Â-area condition guarantees nonvanishing family indices, enabling the band width results to extend beyond previously known cases.

Abstract

We extend Gromov's conjecture on the sharp width estimate for Riemannian bands with positive scalar curvature to the family case and prove that it holds for fiber bundles with infinite family A-hat area. The method we employ is based on Dirac operators and the family index theory. Our proof relies on a product formula for index bundles established in this paper.

Paper Structure

This paper contains 9 sections, 10 theorems, 142 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Index Bundle and Family of twisted Dirac Operators
  3. Index Bundle for Noncompact Fibers
  4. Twisted Spinor Bundles and Families of Twisted Dirac Operators
  5. Family Product Formula
  6. The Case of Even-Dimensional $Z$
  7. The Case of Odd-Dimensional $Z$
  8. Family Band Width Estimate
  9. Proofs of Lemma \ref{['IndBunLem']} and Lemma \ref{["IndBunLem'"]}

Key Result

Theorem 6

Let $N\to B$ be a fiber bundle with infinite family $\widehat{A}$-area. Denote the fiber over $b\in B$ by $Z_b$, which is diffeomorphic to $Z$ for all $b$. Consider $M\coloneq N\times[-1,1]$, which clearly admits a fiber bundle structure over $B$ with fiber $Z\times[-1,1]$. Given a Riemannian metric where $n\coloneq\dim Z+1$.

Figures (12)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 5:
  • ...and 7 more figures

Theorems & Definitions (26)

  • Conjecture 1
  • Conjecture 2
  • Remark 3
  • Example 4
  • Remark 5
  • Theorem 6
  • Definition 1.1
  • Proposition 1.2
  • Proposition 2.1
  • Lemma 2.2
  • ...and 16 more