Characteristic classes of bundles of K3 manifolds and the Nielsen realization problem

pdf

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)→pi_0Diff(K) does not split (even virtually). 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.