Cobordism invariance and the well-definedness of local index

TL;DR

This paper proves the cobordism invariance of the analytic index for Dirac-type operators on open manifolds and demonstrates that the index is well-defined regardless of the open covering chosen.

Contribution

It establishes cobordism invariance of the index and confirms the index's independence from the open covering, advancing the understanding of index theory on open manifolds.

Findings

Proves cobordism invariance of the index.

Shows the index is well-defined independent of open covering.

Extends the analytic index theory to open manifolds.

Abstract

In the previous papers, Furuta, Yoshida and the author gave a definition of analytic index theory of Dirac-type operator on open manifolds by making use of some geometric structure on an open covering of the end of the open manifold and a perturbation of the Dirac-type operator. In this paper we show the cobordism invariance of the index, and as an application we show the well-definedness of the index with respect to the choice of the open covering.