A Product Formula for Family Indices and Family Band Width Estimates

TL;DR

This paper extends Gromov's conjecture on width estimates to families of Riemannian bands with positive scalar curvature, using Dirac operators and a new product formula for index bundles.

Contribution

It introduces a product formula for index bundles and applies it to prove a family version of Gromov's width conjecture for fiber bundles with infinite family A-hat area.

Findings

Proved the family width estimate conjecture for certain fiber bundles.

Established a new product formula for index bundles.

Extended scalar curvature width estimates to the family setting.

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.