Timeline for What's the sufficient or necessary conditions for a manifold to have Lie group structure?
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| when toggle format | what | by | license | comment | |
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| Sep 11, 2017 at 8:51 | answer | added | John Rognes | timeline score: 11 | |
| Sep 11, 2017 at 8:24 | comment | added | YCor | Another remark about $\pi_3$: if a connected manifold carries a Lie group structure and has trivial $\pi_3$ then it's a torus. | |
| Sep 11, 2017 at 8:21 | comment | added | YCor | I'll understand the question in the smooth setting. 1) A manifold has a Lie group structure iff all its components are diffeomorphic and if some of its component admits a Lie group structure. This reduces to the connected case. 2) A connected manifold has a Lie group structure iff it's diffeomorphic to $K\times\mathbf{R}^n$ for some $n$ and some compact Lie group $K$. This reduces to the connected compact case, for which there is a full classification. | |
| Sep 11, 2017 at 7:53 | comment | added | Ali Taghavi | A related post could be the following mathoverflow.net/questions/5262/lie-groups-and-manifolds/… | |
| Sep 11, 2017 at 7:38 | comment | added | Benoît Kloeckner | Of course a Lie group is parallelisable. | |
| Sep 11, 2017 at 6:09 | comment | added | Mark Grant | I can think of a couple more necessary conditions: the second homotopy group must be trivial, and the third torsion-free, according to mathoverflow.net/questions/8957/… . Also, the (co)homology with field coefficients must carry a Hopf algebra structure. | |
| Sep 11, 2017 at 5:16 | history | asked | fff123123 | CC BY-SA 3.0 |