Higher analytic torsion and cohomology of diffeomorphism groups

TL;DR

This paper explores the invariance of higher analytic torsion forms under diffeomorphisms of manifolds, constructs new cohomology classes for diffeomorphism groups, and computes these classes explicitly for the circle, revealing non-trivial group cohomology.

Contribution

It introduces a method to define continuous cohomology classes of diffeomorphism groups using higher analytic torsion forms and computes these classes explicitly for the circle.

Findings

Higher analytic torsion forms are invariant under the identity component of the diffeomorphism group.

Constructed new continuous cohomology classes of diffeomorphism groups from torsion forms.

For the circle, the group cohomology class in degree 2 is non-trivial, while the Lie algebra class vanishes.

Abstract

We consider a closed odd-dimensional oriented manifold $M$ together with an acyclic flat hermitean vector bundle $\cF$. We form the trivial fibre bundle with fibre $M$ over the manifold of all Riemannian metrics on $M$. It has a natural flat connection and a vertical Riemannian metric. The higher analytic torsion form of Bismut/Lott associated to the situation is invariant with respect to the connected component of the identity of the diffeomorphism group of $M$. Using that the space of Riemannian metrics is contractible we define continuous cohomology classes of the diffeomorphism group and its Lie algebra. For the circle we compute this classes in degree 2 and show that the group cohomology class is non-trivial, while the Lie algebra cohomology class vanishes.