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

Hokuto Konno

Algebraic & Geometric Topology 24 (2024) 1739–1753
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 K3 and S2 × S2. The main ingredients of the proofs are Y Kato’s 108–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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