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 M) to BDiff(M).