Connected sums of codimension two locally flat submanifolds
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866912302472626176 |
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| author | Livingston, Charles |
| author_facet | Livingston, Charles |
| contents | Let X and Y be oriented topological manifolds of dimension n + 2, and let K and J be connected, locally-flat, oriented, n-dimensional submanifolds of X and Y. We show that up to orientation preserving homeomorphism there is a well-defined connected sum K # J in X # Y. For n = 1, the proof is classical, relying on results of Rado and Moise. For dimensions n = 3 and n > 5, results of Edwards-Kirby, Kirby, and Kirby-Siebenmann concerning higher dimensional topological manifolds are required. For n = 2, 4, and 5, Freedman and Quinn's work on topological four-manifolds is needed. The truth of the corresponding statement for higher codimension seems to be unknown. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_03930 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Connected sums of codimension two locally flat submanifolds Livingston, Charles Geometric Topology 57N35 Let X and Y be oriented topological manifolds of dimension n + 2, and let K and J be connected, locally-flat, oriented, n-dimensional submanifolds of X and Y. We show that up to orientation preserving homeomorphism there is a well-defined connected sum K # J in X # Y. For n = 1, the proof is classical, relying on results of Rado and Moise. For dimensions n = 3 and n > 5, results of Edwards-Kirby, Kirby, and Kirby-Siebenmann concerning higher dimensional topological manifolds are required. For n = 2, 4, and 5, Freedman and Quinn's work on topological four-manifolds is needed. The truth of the corresponding statement for higher codimension seems to be unknown. |
| title | Connected sums of codimension two locally flat submanifolds |
| topic | Geometric Topology 57N35 |
| url | https://arxiv.org/abs/2104.03930 |