Torus fibrations and localization of index II

TL;DR

This paper develops a localization framework for the index of Dirac-type operators on open manifolds with torus bundle structures, demonstrating properties like excision and product formulas, and showing the index localizes on a compact subset.

Contribution

It introduces a new localization method for Dirac-type operator indices on open manifolds with torus fibrations, utilizing Witten deformation and establishing key properties.

Findings

Index is localized on the compact set

Excision property holds for the index

Product formula for the index is established

Abstract

We give a framework of localization for the index of a Dirac-type operator on an open manifold. Suppose the open manifold has a compact subset whose complement is covered by a family of finitely many open subsets, each of which has a structure of the total space of a torus bundle. Under an acyclic condition we define the index of the Dirac-type operator by using the Witten-type deformation, and show that the index has several properties, such as excision property and a product formula. In particular, we show that the index is localized on the compact set.