Timeline for Examples of interesting non orientable closed 3-manifolds
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Sep 28 at 19:18 | vote | accept | coudy | ||
| S Jun 9, 2018 at 11:42 | history | suggested | Ali Taghavi |
I add a tag.
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| Jun 9, 2018 at 10:06 | review | Suggested edits | |||
| S Jun 9, 2018 at 11:42 | |||||
| Sep 20, 2017 at 15:19 | comment | added | Nick L | One contrast with the orientable case is that non-orientable closed 3-manifolds always have infinite $\pi_{1}$. See here math.stackexchange.com/questions/421303/…. | |
| Sep 20, 2017 at 15:03 | answer | added | Benoît Kloeckner | timeline score: 11 | |
| Sep 20, 2017 at 14:43 | answer | added | Ian Agol | timeline score: 23 | |
| Sep 20, 2017 at 14:16 | comment | added | Arun Debray | Thus even a classification of homotopy classes of closed, unoriented 3-manifolds is likely to be complicated. | |
| Sep 20, 2017 at 14:15 | comment | added | Arun Debray | There is a theorem of Postnikov that if $V$ is a finite-dimensional $\mathbb F_2$-vector space, $\theta$ is a trilinear form on $V$, and $w\in V$ is such that $\theta(w, w, v) = \theta(w, v, v)$ for all $v$, then there's a closed 3-manifold $M$ and an isomorphism $H^1(M;\mathbb F_2)\cong V$ carrying $w_1(M)\mapsto w$ and the cup product $(a, b,c)\mapsto a\smile b\smile c$ to $\theta$. The upshot is that as soon as you've satisfied the Wu formula and Poincaré duality, you have complete freedom in choosing the mod 2 cohomology ring of a closed 3-manifold, and if $w_1\ne 0$, it's unorientable. | |
| Sep 20, 2017 at 13:54 | history | edited | coudy | CC BY-SA 3.0 |
compact -> closed
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| Sep 20, 2017 at 13:53 | comment | added | coudy | Yes indeed, I am interested in boundaryless manifolds - edited. | |
| Sep 20, 2017 at 13:39 | comment | added | Peter Heinig | Is only assuming 'compact' important for you? If not, I think it would be good to make it 'non-orientable 3-dimensional closed manifold'. Then, I think, people knowing much about Thurston's geometrization conjecture could, hopefully, relate your question to the current state of knowledge about geometrization. (For compact yet non-closed manifolds there is, I think, 'less' uniqueness in the geometric model structures, though much is understood for compact non-closed manifolds as well.) I am just suggesting you place your question 'squarely' into the best-understood context of all. | |
| Sep 20, 2017 at 13:16 | history | asked | coudy | CC BY-SA 3.0 |