Pseudo-isotopies of simply connected 4-manifolds
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866911532979322880 |
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| author | Gabai, David Gay, David T. Hartman, Daniel Krushkal, Vyacheslav Powell, Mark |
| author_facet | Gabai, David Gay, David T. Hartman, Daniel Krushkal, Vyacheslav Powell, Mark |
| contents | Perron and Quinn gave independent proofs in 1986 that every topological pseudo-isotopy of a simply-connected, compact topological 4-manifold is isotopic to the identity. Another result of Quinn is that every smooth pseudo-isotopy of a simply-connected, compact, smooth 4-manifold is smoothly stably isotopic to the identity. From this he deduced that $π_4(\operatorname{TOP}(4)/\operatorname{O}(4)) =0$. A replacement criterion is used at a key juncture in Quinn's proofs, but the justification given for it is incorrect. We provide different arguments that bypass the replacement criterion, thus completing Quinn's proofs of both the topological and the stable smooth pseudo-isotopy theorems. We discuss the replacement criterion and state it as an open problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_11196 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Pseudo-isotopies of simply connected 4-manifolds Gabai, David Gay, David T. Hartman, Daniel Krushkal, Vyacheslav Powell, Mark Geometric Topology 57K40, 57N35 Perron and Quinn gave independent proofs in 1986 that every topological pseudo-isotopy of a simply-connected, compact topological 4-manifold is isotopic to the identity. Another result of Quinn is that every smooth pseudo-isotopy of a simply-connected, compact, smooth 4-manifold is smoothly stably isotopic to the identity. From this he deduced that $π_4(\operatorname{TOP}(4)/\operatorname{O}(4)) =0$. A replacement criterion is used at a key juncture in Quinn's proofs, but the justification given for it is incorrect. We provide different arguments that bypass the replacement criterion, thus completing Quinn's proofs of both the topological and the stable smooth pseudo-isotopy theorems. We discuss the replacement criterion and state it as an open problem. |
| title | Pseudo-isotopies of simply connected 4-manifolds |
| topic | Geometric Topology 57K40, 57N35 |
| url | https://arxiv.org/abs/2311.11196 |