Characteristic classes of bundles of K3 manifolds and the Nielsen
Let $K$ be the K3 manifold. In this note, we discuss two methods to prove
that certain generalized Miller--Morita--Mumford classes for smooth bundles
with fiber $K$ are non-zero. As a consequence, we fill a gap in a paper of the
first author, and prove that the homomorphism $Diff(K)\to \pi_0 Diff(K)$ does
not split. One of the two methods of proof uses a result of Franke on the
stable cohomology of arithmetic groups that strengthens work of Borel, and may
be of independent interest.