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

Konno, Hokuto

doi:10.2140/agt.2024.24.1739

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Dehn twists and the Nielsen realization problem for spin $4$–manifolds

Hokuto Konno

Algebraic & Geometric Topology 24 (2024) 1739–1753

DOI: 10.2140/agt.2024.24.1739

Abstract

We prove that for a closed oriented smooth spin $4$ –manifold $X$ with nonzero 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 $K 3$ 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.

Keywords

diffeomorphism group, 4–manifold, group action

Mathematical Subject Classification

Primary: 57S17

References

Publication

Received: 27 July 2022

Revised: 24 December 2022

Accepted: 2 February 2023

Published: 28 June 2024

Authors

Hokuto Konno
Graduate School of Mathematical Sciences The University of Tokyo Tokyo Japan

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Volume 24, issue 3 (2024)