Non-metrizable manifolds and contractibility
We investigate whether non-metrizable manifolds in various classes can be homotopy equivalent to a CW-complex (in short: heCWc), and in particular contractible. We show that a non-metrizable manifold cannot be heCWc if it has one of the following properties: it contains a countably compact non-compa...
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| Language: | English |
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Universitat Politècnica de València
2025-04-01
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| Series: | Applied General Topology |
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| Online Access: | https://polipapers.upv.es/index.php/AGT/article/view/21796 |
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| author | Mathieu Baillif |
| author_facet | Mathieu Baillif |
| author_sort | Mathieu Baillif |
| collection | DOAJ |
| description | We investigate whether non-metrizable manifolds in various classes can be homotopy equivalent to a CW-complex (in short: heCWc),
and in particular contractible. We show that a non-metrizable manifold cannot be heCWc if it has one of the following properties: it contains a countably compact non-compact subspace; it contains a copy of an ω1-compact subspace of an ω1-tree; it contains a non-Lindelöf closed subspace functionally narrow in it. (These results hold for more general spaces than just manifolds.) We also show that the positive part of the tangent bundle of the longray is never heCWc for any smoothing. These theorems follow from stabilization properties of real valued maps. On a more geometric side, we also show that the Prüfer surface, which has been shown to be contractible long ago, has an open submanifold with trivial homotopy groups which is not heCWc. On the other end of the spectrum, we show that there is a non-metrizable contractible Type I surface. |
| format | Article |
| id | doaj-art-1d1f050ca0a8479fb656c0604afe116d |
| institution | OA Journals |
| issn | 1576-9402 1989-4147 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | Universitat Politècnica de València |
| record_format | Article |
| series | Applied General Topology |
| spelling | doaj-art-1d1f050ca0a8479fb656c0604afe116d2025-08-20T01:50:50ZengUniversitat Politècnica de ValènciaApplied General Topology1576-94021989-41472025-04-0126130333910.4995/agt.2025.2179620983Non-metrizable manifolds and contractibilityMathieu Baillif0HES-SO GenèveWe investigate whether non-metrizable manifolds in various classes can be homotopy equivalent to a CW-complex (in short: heCWc), and in particular contractible. We show that a non-metrizable manifold cannot be heCWc if it has one of the following properties: it contains a countably compact non-compact subspace; it contains a copy of an ω1-compact subspace of an ω1-tree; it contains a non-Lindelöf closed subspace functionally narrow in it. (These results hold for more general spaces than just manifolds.) We also show that the positive part of the tangent bundle of the longray is never heCWc for any smoothing. These theorems follow from stabilization properties of real valued maps. On a more geometric side, we also show that the Prüfer surface, which has been shown to be contractible long ago, has an open submanifold with trivial homotopy groups which is not heCWc. On the other end of the spectrum, we show that there is a non-metrizable contractible Type I surface.https://polipapers.upv.es/index.php/AGT/article/view/21796non-metrizable manifoldscountably compact manifoldsroad space of treescontractible spaces |
| spellingShingle | Mathieu Baillif Non-metrizable manifolds and contractibility Applied General Topology non-metrizable manifolds countably compact manifolds road space of trees contractible spaces |
| title | Non-metrizable manifolds and contractibility |
| title_full | Non-metrizable manifolds and contractibility |
| title_fullStr | Non-metrizable manifolds and contractibility |
| title_full_unstemmed | Non-metrizable manifolds and contractibility |
| title_short | Non-metrizable manifolds and contractibility |
| title_sort | non metrizable manifolds and contractibility |
| topic | non-metrizable manifolds countably compact manifolds road space of trees contractible spaces |
| url | https://polipapers.upv.es/index.php/AGT/article/view/21796 |
| work_keys_str_mv | AT mathieubaillif nonmetrizablemanifoldsandcontractibility |