# A topological approach to indices of geometric operators on manifolds with fibered boundaries

## TL;DR

This paper explores the topological properties of indices of twisted geometric operators on manifolds with fibered boundaries, introducing new $K$-groups and applying groupoid deformation techniques to analyze their behavior.

## Contribution

It defines $K$-groups relative to boundary fibrations and demonstrates how indices of twisted operators relate to these groups, advancing the understanding of geometric operator indices on fibered boundary manifolds.

## Key findings

- Indices can be viewed as pairings over new $K$-groups.
- Groupoid deformation techniques reveal properties of these indices.
- Application to signature operator localization on singular fiber bundles.

## Abstract

In this paper, we investigate topological aspects of indices of twisted geometric operators on manifolds equipped with fibered boundaries. We define $K$-groups relative to the pushforward for boundary fibration, and show that indices of twisted geometric operators, defined by complete $\Phi$ or edge metrics, can be regarded as the index pairing over these $K$-groups. We also prove various properties of these indices using groupoid deformation techniques. Using these properties, we give an application to the localization problem of signature operators for singular fiber bundles.