A 4-dimensional pseudo-Anosov homeomorphism
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908650676682752 |
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| author | Martelli, Bruno |
| author_facet | Martelli, Bruno |
| contents | We know from previous work with Italiano and Migliorini that there exists some hyperbolic 5-manifold that fibers over the circle. Here we build one example where the monodromy is a "pseudo-Anosov homeomorphism" of the 4-dimensional fiber, in a way that is surprisingly similar to the familiar and beautiful two-dimensional picture of Nielsen and Thurston for surfaces. This fact has various consequences:
(1) There is a compact smooth 4-manifold $M$ such that no non-trivial class in $H_2(M)$ is represented by immersed tori, and infinitely many classes are represented by smoothly embedded genus two surfaces.
(2) There is a compact locally CAT(0) space $Y$ such that $π_1(Y)$ is not hyperbolic and does not contain $\mathbb Z \times \mathbb Z$.
The latter answers a question of Gromov, known as the Closing Flat Problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_10530 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A 4-dimensional pseudo-Anosov homeomorphism Martelli, Bruno Geometric Topology Differential Geometry Dynamical Systems We know from previous work with Italiano and Migliorini that there exists some hyperbolic 5-manifold that fibers over the circle. Here we build one example where the monodromy is a "pseudo-Anosov homeomorphism" of the 4-dimensional fiber, in a way that is surprisingly similar to the familiar and beautiful two-dimensional picture of Nielsen and Thurston for surfaces. This fact has various consequences: (1) There is a compact smooth 4-manifold $M$ such that no non-trivial class in $H_2(M)$ is represented by immersed tori, and infinitely many classes are represented by smoothly embedded genus two surfaces. (2) There is a compact locally CAT(0) space $Y$ such that $π_1(Y)$ is not hyperbolic and does not contain $\mathbb Z \times \mathbb Z$. The latter answers a question of Gromov, known as the Closing Flat Problem. |
| title | A 4-dimensional pseudo-Anosov homeomorphism |
| topic | Geometric Topology Differential Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2511.10530 |