The Casson-Sullivan invariant for homeomorphisms of 4-manifolds
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916244231290880 |
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| author | Galvin, Daniel A. P. |
| author_facet | Galvin, Daniel A. P. |
| contents | We investigate the realisability of the Casson-Sullivan invariant for homeomorphisms of smooth $4$-manifolds, which is the obstruction to a homeomorphism being stably pseudo-isotopic to a diffeomorphism, valued in the third cohomology of the source manifold with $\mathbb{Z}/2$-coefficients. We prove that for all orientable pairs of homeomorphic, smooth $4$-manifolds this invariant can be realised fully after stabilising with a single $S^2\times S^2$. As an application, we obtain that topologically isotopic surfaces in a smooth, simply-connected $4$-manifold become smoothly isotopic after sufficient external stabilisations. We further demonstrate cases where this invariant can be realised fully without stabilisation for self-homeomorphisms, which includes for manifolds with finite cyclic fundamental group. This method allows us to produce many examples of homeomorphisms which are not stably pseudo-isotopic to any diffeomorphism but are homotopic to the identity. Finally, we reinterpret these results in terms of finding examples of smooth structures on $4$-manifolds which are diffeomorphic but not stably pseudo-isotopic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_07928 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Casson-Sullivan invariant for homeomorphisms of 4-manifolds Galvin, Daniel A. P. Geometric Topology 57K40, 57R10 (Primary) 57K10, 57N37, 57R50, 57R65, (Secondary) We investigate the realisability of the Casson-Sullivan invariant for homeomorphisms of smooth $4$-manifolds, which is the obstruction to a homeomorphism being stably pseudo-isotopic to a diffeomorphism, valued in the third cohomology of the source manifold with $\mathbb{Z}/2$-coefficients. We prove that for all orientable pairs of homeomorphic, smooth $4$-manifolds this invariant can be realised fully after stabilising with a single $S^2\times S^2$. As an application, we obtain that topologically isotopic surfaces in a smooth, simply-connected $4$-manifold become smoothly isotopic after sufficient external stabilisations. We further demonstrate cases where this invariant can be realised fully without stabilisation for self-homeomorphisms, which includes for manifolds with finite cyclic fundamental group. This method allows us to produce many examples of homeomorphisms which are not stably pseudo-isotopic to any diffeomorphism but are homotopic to the identity. Finally, we reinterpret these results in terms of finding examples of smooth structures on $4$-manifolds which are diffeomorphic but not stably pseudo-isotopic. |
| title | The Casson-Sullivan invariant for homeomorphisms of 4-manifolds |
| topic | Geometric Topology 57K40, 57R10 (Primary) 57K10, 57N37, 57R50, 57R65, (Secondary) |
| url | https://arxiv.org/abs/2405.07928 |