Diffeomorphisms, Analytic Torsion and Noncommutative Geometry

TL;DR

This paper establishes an index theorem for flat B-vector bundles, constructs an associated analytic torsion form, and demonstrates its use as a homotopy invariant for the diffeomorphism group of aspherical manifolds.

Contribution

It introduces a new index theorem for flat B-vector bundles and constructs an analytic torsion form that detects homotopy invariants of diffeomorphism groups.

Findings

Proved an index theorem for pushforward of flat B-vector bundles.

Constructed an analytic torsion form T associated with these bundles.

Showed T provides invariants of the homotopy groups of Diff(Z) for aspherical manifolds.

Abstract

We prove an index theorem concerning the pushforward of flat B-vector bundles, where B is an appropriate algebra. We construct the associated analytic torsion form T. If Z is a smooth closed aspherical manifold, we show that T gives invariants of the homotopy groups of Diff(Z).