Dehn twists and the Nielsen realization problem for spin 4-manifolds

TL;DR

This paper demonstrates that certain Dehn twists in spin 4-manifolds with non-zero signature cannot be realized as finite order diffeomorphisms, providing a negative answer to the Nielsen realization problem in this context.

Contribution

It proves the non-realizability of Dehn twists as finite order diffeomorphisms in specific spin 4-manifolds and highlights differences between topological and smooth categories.

Findings

Dehn twists about (+2) or (-2)-spheres are not homotopic to finite order diffeomorphisms in certain spin 4-manifolds.

Negative solution to the Nielsen realization problem for groups generated by Dehn twists.

Discrepancy between topological and smooth Nielsen realization problems for connected sums of K3 and S^2×S^2.

Abstract

We prove that, for a closed oriented smooth spin 4-manifold $X$ with non-zero signature, the Dehn twist about a $(+2)$- or $(-2)$-sphere in $X$ is not homotopic to any finite order diffeomorphism. In particular, we negatively answer the Nielsen realization problem for each group generated by the mapping class of a Dehn twist. We also show that there is a discrepancy between the Nielsen realization problems in the topological category and smooth category for connected sums of copies of $K3$ and $S^{2} \times S^{2}$. The main ingredients of the proofs are Y. Kato's 10/8-type inequality for involutions and a refinement of it.