The diffeomorphism group of a K3 surface and Nielsen realization

TL;DR

This paper proves that for certain 4-manifolds containing a K3 surface, there is no section for the Nielsen realization problem over the entire mapping class group, using obstructions in rational cohomology.

Contribution

It establishes the first nonexistence theorem in dimension 4 for the Nielsen realization problem involving K3 surfaces.

Findings

Obstructions in rational cohomology are nonzero for these manifolds.

No section exists over the entire mapping class group for manifolds with a K3 summand.

The obstructions are detected via the moduli space of Einstein metrics.

Abstract

The Nielsen Realization problem asks when the group homomorphism from Diff(M) to pi_0 Diff(M) admits a section. For M a closed surface, Kerckhoff proved that a section exists over any finite subgroup, but Morita proved that if the genus is large enough then no section exists over the entire mapping class group. We prove the first nonexistence theorem of this type in dimension 4: if M is a smooth closed oriented 4-manifold which contains a K3 surface as a connected summand then no section exists over the whole of the mapping class group. This is done by showing that certain obstructions lying in the rational cohomology of B(pi_0 Diff(M)) are nonzero. We detect these classes by showing that they are nonzero when pulled back to the moduli space of Einstein metrics on a K3 surface.