Exotic Dehn twists on 4-manifolds
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913291273502720 |
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| author | Konno, Hokuto Mallick, Abhishek Taniguchi, Masaki |
| author_facet | Konno, Hokuto Mallick, Abhishek Taniguchi, Masaki |
| contents | We initiate the study of exotic Dehn twists along 3-manifolds $\neq S^3$ inside $4$-manifolds, which produces the first known examples of exotic diffeomorphisms of contractible 4-manifolds, more generally of definite 4-manifolds, and exotic diffeomorphisms of 4-manifolds with $\neq S^3$ boundary that survive after one stabilization. We also construct the smallest closed 4-manifold known to support an exotic diffeomorphism. These exotic diffeomorphisms are the Dehn twists along certain Seifert fibered 3-manifolds. As a consequence, we get loops of diffeomorphisms of 3-manifolds that topologically extend to some 4-manifolds $X$ but not smoothly so, implying the non-surjectivity of $π_1(\mathrm{Diff}(X)) \to π_1(\mathrm{Homeo}(X))$. Our method uses $2$-parameter families Seiberg-Witten theory over $\mathbb{RP}^2$, while known methods to detect exotic diffeomorphisms used $1$-parameter families gauge-theoretic invariants. Using a similar strategy, we construct a new kind of exotic diffeomorphisms of 4-manifolds, given as commutators of diffeomorphisms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_08607 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Exotic Dehn twists on 4-manifolds Konno, Hokuto Mallick, Abhishek Taniguchi, Masaki Geometric Topology Differential Geometry We initiate the study of exotic Dehn twists along 3-manifolds $\neq S^3$ inside $4$-manifolds, which produces the first known examples of exotic diffeomorphisms of contractible 4-manifolds, more generally of definite 4-manifolds, and exotic diffeomorphisms of 4-manifolds with $\neq S^3$ boundary that survive after one stabilization. We also construct the smallest closed 4-manifold known to support an exotic diffeomorphism. These exotic diffeomorphisms are the Dehn twists along certain Seifert fibered 3-manifolds. As a consequence, we get loops of diffeomorphisms of 3-manifolds that topologically extend to some 4-manifolds $X$ but not smoothly so, implying the non-surjectivity of $π_1(\mathrm{Diff}(X)) \to π_1(\mathrm{Homeo}(X))$. Our method uses $2$-parameter families Seiberg-Witten theory over $\mathbb{RP}^2$, while known methods to detect exotic diffeomorphisms used $1$-parameter families gauge-theoretic invariants. Using a similar strategy, we construct a new kind of exotic diffeomorphisms of 4-manifolds, given as commutators of diffeomorphisms. |
| title | Exotic Dehn twists on 4-manifolds |
| topic | Geometric Topology Differential Geometry |
| url | https://arxiv.org/abs/2306.08607 |