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

## TL;DR

This paper demonstrates that certain characteristic classes for smooth K3 fiber bundles are non-zero, filling a gap in previous work and showing the diffeomorphism group of K3 surfaces does not split, using advanced cohomological methods.

## Contribution

It introduces two methods to prove non-vanishing of Miller--Morita--Mumford classes for K3 bundles and establishes that the diffeomorphism group of K3 manifolds does not split.

## Key findings

- Certain characteristic classes are non-zero for K3 bundles.
- The homomorphism from the diffeomorphism group to its isotopy classes does not split.
- A new proof technique involving stable cohomology of arithmetic groups.

## Abstract

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.