Cobordism invariance of the family index

TL;DR

This paper proves that the family index remains invariant under cobordism using K-theory, focusing on elliptic pseudodifferential families over fiber bundles and employing push-forward maps.

Contribution

It provides a K-theory proof of cobordism invariance of the family index for elliptic pseudodifferential operators on fiber bundles.

Findings

Index is invariant under cobordism of families.

Uses K-theory and push-forward maps to establish invariance.

Reduces problem to families on B x R^n.

Abstract

We give a K-theory proof of the invariance under cobordism of the family index. We consider elliptic pseudodifferential families on a continuous fibre bundle with smooth fibres over a compact base space B, and define a notion of cobordant families using K^1-groups on fibrations with boundary. We show that the index of two such families is the same using properties of the push-forward map in K-theory to reduce it to families on B x R^n.