Ends and end cohomology
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910912007372800 |
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| author | Bass, William G. Calcut, Jack S. |
| author_facet | Bass, William G. Calcut, Jack S. |
| contents | Ends and end cohomology are powerful invariants for the study of noncompact spaces. We present a self-contained exposition of the topological theory of ends and prove novel extensions including the existence of an exhaustion of a proper map. We define reduced end cohomology as the relative end cohomology of a ray-based space. We use those results to prove a version of a theorem of King that computes the reduced end cohomology of an end sum of two manifolds. We include a complete proof of Freudenthal's fundamental theorem on the number of ends of a topological group, and we use our results on dimension-zero end cohomology to prove -- without using transfinite induction -- a theorem of Nöbeling on freeness of certain modules of continuous functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_01816 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ends and end cohomology Bass, William G. Calcut, Jack S. Algebraic Topology General Topology Geometric Topology 54D35, 55N20, 55P57 (Primary) 54H11 (Secondary) Ends and end cohomology are powerful invariants for the study of noncompact spaces. We present a self-contained exposition of the topological theory of ends and prove novel extensions including the existence of an exhaustion of a proper map. We define reduced end cohomology as the relative end cohomology of a ray-based space. We use those results to prove a version of a theorem of King that computes the reduced end cohomology of an end sum of two manifolds. We include a complete proof of Freudenthal's fundamental theorem on the number of ends of a topological group, and we use our results on dimension-zero end cohomology to prove -- without using transfinite induction -- a theorem of Nöbeling on freeness of certain modules of continuous functions. |
| title | Ends and end cohomology |
| topic | Algebraic Topology General Topology Geometric Topology 54D35, 55N20, 55P57 (Primary) 54H11 (Secondary) |
| url | https://arxiv.org/abs/2412.01816 |