The diffeomorphism group of a K3 surface and Nielsen realization

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.

20 pages, published version. ERRATUM: This paper is withdrawn. As pointed out by Bena Tshishiku, Borel's theorem on the cohomology stable range for arithmetic groups is incorrectly quoted and applied in this paper; in the case relevant to K3 surfaces the range is unfortunately zero. Hence the proof of the part of Theorem 1.1 referring to K3 surfaces is fundamentally broken. See arXiv:1711.03139

References in corpus (1)

• Characteristic classes of smooth fibrations

Cited by in corpus (7)

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• Characteristic classes of bundles of K3 manifolds and the Nielsen realization problem
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• Algebraic independence of generalized Morita-Miller-Mumford classes
• Dehn twists and the Nielsen realization problem for spin 4-manifolds
• Cohomological obstructions to Nielsen realization
• Non-trivial smooth families of $K3$ surfaces