An index theorem for manifolds with boundary

TL;DR

This paper extends the Atiyah-Singer index theorem to manifolds with boundary by utilizing deformation groupoids and their actions, building on Connes' approach for closed manifolds.

Contribution

It introduces a new index theorem for manifolds with boundary, adapting Connes' deformation groupoid methods to this setting.

Findings

Proves an index theorem for manifolds with boundary.

Uses deformation groupoids to handle boundary conditions.

Extends Connes' approach from closed to bounded manifolds.

Abstract

In his book (II.5), Connes gives a proof of the Atiyah-Singer index theorem for closed manifolds by using deformation groupoids and appropiate actions of these on R^N. Following these ideas, we prove an index theorem for manifolds with boundary.