On the mapping class groups of simply connected smooth 4-manifolds

Baraglia, David

doi:10.2140/agt.2026.26.1635

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On the mapping class groups of simply connected smooth $4$-manifolds

David Baraglia

Algebraic & Geometric Topology 26 (2026) 1635–1653

DOI: 10.2140/agt.2026.26.1635

Abstract

The mapping class group $M (X)$ of a smooth manifold $X$ is the group of smooth isotopy classes of orientation-preserving diffeomorphisms of $X$ . We prove a number of results about the mapping class groups of compact, simply connected, smooth $4$ -manifolds. For example, we prove that $M (X)$ is nonfinitely generated for $X = 2 n {ℂ ℙ}^{2} # 10 n \bar{{ℂ ℙ}^{2}}$ , where $n \geq 3$ is odd. Let $Γ (X)$ denote the group of automorphisms of the intersection lattice of $X$ that can be realised by diffeomorphisms. Then $M (X)$ is an extension of $Γ (X)$ by $T (X)$ , the Torelli group of isotopy classes of diffeomorphisms that act trivially in cohomology. We prove this extension is split for connected sums of ${ℂ ℙ}^{2}$ , but is not split for $2 {ℂ ℙ}^{2} # n \bar{{ℂ ℙ}^{2}}$ , where $n \geq 11$ . We prove that the Nielsen realisation problem fails for certain finite subgroups of $M (p {ℂ ℙ}^{2} # q \bar{{ℂ ℙ}^{2}})$ whenever $p + q \geq 4$ . Lastly we study the extension $M_{1} (X) \to M (X)$ , where $M_{1} (X)$ is the group of isotopy classes of diffeomorphisms of $X$ which fix a neighbourhood of a point. When $X = K 3$ or $K 3 # (S^{2} \times S^{2})$ we prove that $M_{1} (X) \to M (X)$ is a nontrivial extension of $M (X)$ by $ℤ_{2}$ . Moreover, we completely determine the extension class of $M_{1} (K 3) \to M (K 3)$ .

Keywords

Seiberg–Witten, $4$-manifolds, mapping class group

Mathematical Subject Classification

Primary: 57K41, 57R50

References

Publication

Received: 6 November 2023

Revised: 16 March 2025

Accepted: 29 May 2025

Published: 23 May 2026

Authors

David Baraglia
School of Mathematical Sciences The University of Adelaide Adelaide Australia

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Volume 26, issue 5 (2026)