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Exotic Dehn twists on
$4$-manifolds
Hokuto Konno, Abhishek Mallick and Masaki Taniguchi |
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Geometry & Topology 30 (2026) 2395–2429
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DOI: 10.2140/gt.2026.30.2395
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Abstract |
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We initiate the study of exotic Dehn twists along 3-manifolds embedded in 4-manifolds. This yields the first known examples of exotic diffeomorphisms of contractible 4-manifolds; more generally, it produces exotic diffeomorphisms of definite 4-manifolds, as well as exotic diffeomorphisms of 4-manifolds (whose boundary is not diffeomorphic to ) that persist after one stabilization. In particular, we obtain the first example of a diffeomorphism of a contractible 4-manifold that is not isotopic to the identity. We also construct the smallest closed 4-manifold currently known to support an exotic diffeomorphism. These exotic diffeomorphisms arise as Dehn twists along certain Seifert fibered 3-manifolds. As a consequence, we obtain loops of diffeomorphisms of 3-manifolds that extend topologically, but not smoothly, over some 4-manifold . This implies that the map is not surjective. Our method uses Seiberg–Witten theory for a 2-parameter family over , whereas previous methods for detecting exotic diffeomorphisms relied on 1-parameter family gauge-theoretic invariants. Using a similar strategy, we also construct a new type of exotic diffeomorphism of 4-manifolds, realized as commutators of diffeomorphisms. |
Keywords
exotic diffeomorphism, Dehn twists along Seifert
$4$-manifolds, Seiberg–Witten theory for families of
$4$-manifolds, Donaldson's diagonalization theorem for
families
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Mathematical Subject Classification
Primary: 57K41, 57R50, 57R58, 58D05
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References |
Publication
Received: 23 October 2025
Revised: 11 February 2026
Accepted: 12 March 2026
Published: 31 July 2026
Proposed: Ciprian Manolescu
Seconded: David Gabai, András I Stipsicz
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Authors |
| Hokuto Konno | |
| Graduate School of Mathematical
Sciences The University of Tokyo Tokyo Japan |
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| Abhishek Mallick | |
| Department of Mathematics Dartmouth College Hanover, NH United States |
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| Masaki Taniguchi | |
| Department of Mathematics Kyoto University Kyoto Japan |
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| © 2026 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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