Vanishing of universal characteristic classes for handlebody groups and boundary bundles

TL;DR

The paper proves that certain universal characteristic classes vanish in integral cohomology when restricted to handlebody subgroups, extending to all dimensions a general theorem about characteristic classes and diffeomorphism groups.

Contribution

It establishes the vanishing of odd Miller-Morita-Mumford classes for handlebody groups and generalizes this to all dimensions for classes from Pontrjagin and Euler classes.

Findings

Odd Miller-Morita-Mumford classes vanish on handlebody subgroups.

Universal characteristic classes from Pontrjagin and Euler classes vanish when pulled back from boundary diffeomorphism groups.

Results apply in all dimensions, not just surfaces.

Abstract

Using certain Thom spectra appearing in the study of cobordism categories, we show that the odd half of the Miller-Morita-Mumford classes on the mappping class group of a surface with negative Euler characteristic vanish in integral cohomology when restricted to the handlebody subgroup. This is a special case of a more general theorem valid in all dimensions: universal characteristic classes made from monomials in the Pontrjagin classes (and even powers of the Euler class) vanish when pulled back from BDiff(\partial W) to BDiff(W).