Algorithms in 4-manifold topology
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911180067438592 |
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| author | Bastl, Stefan Burke, Rhuaidi Chatterjee, Rima Dey, Subhankar Durst, Alison Friedl, Stefan Galvin, Daniel Rivas, Alejandro García Hirsch, Tobias Hobohm, Cara Hsueh, Chun-Sheng Kegel, Marc Kern, Frieda Lee, Shun Ming Samuel Löh, Clara Manikandan, Naageswaran Mousseau, Léo Munser, Lars Pencovitch, Mark Perras, Patrick Powell, Mark Quintanilha, José Pedro Schambeck, Lisa Suchodoll, David Tancer, Martin Thiele, Annika Truöl, Paula Uschold, Matthias Veselá, Simona Weiß, Melvin von Wunsch-Rolshoven, Magdalina |
| author_facet | Bastl, Stefan Burke, Rhuaidi Chatterjee, Rima Dey, Subhankar Durst, Alison Friedl, Stefan Galvin, Daniel Rivas, Alejandro García Hirsch, Tobias Hobohm, Cara Hsueh, Chun-Sheng Kegel, Marc Kern, Frieda Lee, Shun Ming Samuel Löh, Clara Manikandan, Naageswaran Mousseau, Léo Munser, Lars Pencovitch, Mark Perras, Patrick Powell, Mark Quintanilha, José Pedro Schambeck, Lisa Suchodoll, David Tancer, Martin Thiele, Annika Truöl, Paula Uschold, Matthias Veselá, Simona Weiß, Melvin von Wunsch-Rolshoven, Magdalina |
| contents | We show that there exists an algorithm that takes as input two closed, simply connected, topological 4-manifolds and decides whether or not these 4-manifolds are homeomorphic. In particular, we explain in detail how closed, simply connected, topological 4-manifolds can be naturally represented by a Kirby diagram consisting only of 2-handles. This representation is used as input for our algorithm. Along the way, we develop an algorithm to compute the Kirby-Siebenmann invariant of a closed, simply connected, topological 4-manifold from any of its Kirby diagrams and describe an algorithm that decides whether or not two intersection forms are isometric.
In a slightly different direction, we discuss the decidability of the stable classification of smooth manifolds with more general fundamental groups. Here we show that there exists an algorithm that takes as input two closed, oriented, smooth 4-manifolds with fundamental groups isomorphic to a finite group with cyclic Sylow 2-subgroup, an infinite cyclic group, or a group of geometric dimension at most 3 (in the latter case we additionally assume that the universal covers of both 4-manifolds are not spin), and decides whether or not these two 4-manifolds are orientation-preserving stably diffeomorphic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08775 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Algorithms in 4-manifold topology Bastl, Stefan Burke, Rhuaidi Chatterjee, Rima Dey, Subhankar Durst, Alison Friedl, Stefan Galvin, Daniel Rivas, Alejandro García Hirsch, Tobias Hobohm, Cara Hsueh, Chun-Sheng Kegel, Marc Kern, Frieda Lee, Shun Ming Samuel Löh, Clara Manikandan, Naageswaran Mousseau, Léo Munser, Lars Pencovitch, Mark Perras, Patrick Powell, Mark Quintanilha, José Pedro Schambeck, Lisa Suchodoll, David Tancer, Martin Thiele, Annika Truöl, Paula Uschold, Matthias Veselá, Simona Weiß, Melvin von Wunsch-Rolshoven, Magdalina Geometric Topology 57K40, 57K10, 57R65 We show that there exists an algorithm that takes as input two closed, simply connected, topological 4-manifolds and decides whether or not these 4-manifolds are homeomorphic. In particular, we explain in detail how closed, simply connected, topological 4-manifolds can be naturally represented by a Kirby diagram consisting only of 2-handles. This representation is used as input for our algorithm. Along the way, we develop an algorithm to compute the Kirby-Siebenmann invariant of a closed, simply connected, topological 4-manifold from any of its Kirby diagrams and describe an algorithm that decides whether or not two intersection forms are isometric. In a slightly different direction, we discuss the decidability of the stable classification of smooth manifolds with more general fundamental groups. Here we show that there exists an algorithm that takes as input two closed, oriented, smooth 4-manifolds with fundamental groups isomorphic to a finite group with cyclic Sylow 2-subgroup, an infinite cyclic group, or a group of geometric dimension at most 3 (in the latter case we additionally assume that the universal covers of both 4-manifolds are not spin), and decides whether or not these two 4-manifolds are orientation-preserving stably diffeomorphic. |
| title | Algorithms in 4-manifold topology |
| topic | Geometric Topology 57K40, 57K10, 57R65 |
| url | https://arxiv.org/abs/2411.08775 |