Higher analytic torsion of sphere bundles and continuous cohomology of $Diff(S^{2n-1})$

TL;DR

This paper constructs a characteristic class for smooth sphere bundles using higher analytic torsion forms and explores its implications for the continuous cohomology of the diffeomorphism group of odd-dimensional spheres.

Contribution

It introduces a new characteristic class for sphere bundles derived from higher analytic torsion forms and links it to continuous cohomology classes of sphere diffeomorphism groups.

Findings

Calculation of the characteristic class for sphere bundles from complex vector bundles

Construction of nontrivial continuous cohomology classes of Diff(S^{2n-1})

Establishment of a connection between analytic torsion and diffeomorphism group cohomology

Abstract

Using the higher analytic torsion form of Bismut and Lott we construct a characteristic class for smooth sphere bundles. We calculate this class in the case where the sphere bundle comes from a complex vector bundle. Related to these characteristic classes we define nontrivial continuous group cohomology classes of the diffeomorphism group of odd dimensional spheres.