A note on the Nielsen realization problem for K3 surfaces

Abstract

We will show the following three theorems on the diffeomorphism and homeomorphism groups of a $K3$ surface. The first theorem is that the natural map ${\pi }_0(Diff(K3))\to Aut(H^2(K3;\mathbb{Z}))$ has a section over its image. The second is that, there exists a subgroup $G$ of ${\pi }_0(Diff(K3))$ of order two over which there is no splitting of the map $Diff(K3)\to {\pi }_0(Diff(K3))$, but there is a splitting of $Homeo(K3)\to {\pi }_0(Homeo(K3))$ over the image of $G$ in ${\pi }_0(Homeo(K3))$, which is non-trivial. The third is that the map ${\pi }_1(Diff(K3))\to {\pi }_1(Homeo(K3))$ is not surjective. Our proof of these results is based on Seiberg-Witten theory and the global Torelli theorem for $K3$ surfaces.

8 pages

References in corpus (4)

• On the Bauer-Furuta and Seiberg-Witten invariants of families of $4$-manifolds
• Constraints on families of smooth 4-manifolds from Bauer-Furuta invariants
• Characteristic classes of bundles of K3 manifolds and the Nielsen realization problem
• Rigidity of the mod 2 families Seiberg-Witten invariants and topology of families of spin 4-manifolds

Cited by in corpus (2)

• On the Bauer-Furuta and Seiberg-Witten invariants of families of $4$-manifolds
• Dehn twists and the Nielsen realization problem for spin 4-manifolds