The relative Mishchenko--Fomenko higher index and almost flat bundles I: The relative Mishchenko--Fomenko index

TL;DR

This paper introduces an alternative definition of the relative Mishchenko--Fomenko higher index, demonstrating its equivalence to existing maps and exploring its relation to amalgamated free product groups and dual indices.

Contribution

It provides a new definition of the relative higher index and proves its equivalence to existing definitions, enhancing understanding of its properties and applications.

Findings

The new relative higher index coincides with existing maps.

The relative higher index relates to amalgamated free product groups.

The dual relative higher index map is rationally surjective under certain conditions.

Abstract

In this paper, the first of two, we introduce an alternative definition of the Chang--Weinberger--Yu relative higher index, which is thought of as a relative analogue of the Mishchenko--Fomenko index pairing. A main result of this paper is that our map coincides with the existing relative higher index maps. We make use of this fact for understanding the relative higher index. First, we relate the relative higher index with the higher index of amalgamated free product groups. Second, we define the dual relative higher index map and show its rational surjectivity under certain assumptions.