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Peer-reviewedCC BYGeometry & Topology

Exotic Dehn twists on 4-manifolds

  • Hokuto Konno1
  • Abhishek Mallick2
  • M. Taniguchi3
  1. 1Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, Japan
  2. 2Department of Mathematics, Dartmouth College, Hanover, NH, United States
  3. 3Department of Mathematics, Kyoto University, Kyoto, Japan
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Abstract

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 S 3 ) 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 X .This implies that the map 1 .Diff.X // ! 1 .Homeo.X // is not surjective.Our method uses Seiberg-Witten theory for a 2-parameter family over RP 2 , 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.

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Cite

BibTeXRIS
BibTeX
@article{konno2026exotic,
  title = {{Exotic Dehn twists on 4-manifolds}},
  author = {Hokuto Konno and Abhishek Mallick and M. Taniguchi},
  journal = {Geometry \& Topology},
  year = {2026},
  month = jul,
  doi = {10.2140/gt.2026.30.2395},
  url = {https://doi.org/10.2140/gt.2026.30.2395},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Exotic Dehn twists on 4-manifolds
AU  - Hokuto Konno
AU  - Abhishek Mallick
AU  - M. Taniguchi
PY  - 2026
DA  - 2026/07/31
JO  - Geometry & Topology
DO  - 10.2140/gt.2026.30.2395
UR  - https://doi.org/10.2140/gt.2026.30.2395
AB  - 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 S 3 ) 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 X .This implies that the map 1 .Diff.X // ! 1 .Homeo.X // is not surjective.Our method uses Seiberg-Witten theory for a 2-parameter family over RP 2 , 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.
SN  - 1364-0380
SN  - 1465-3060
ER  -