Timeline for How to compute second homology of a group given by presentation with two relators
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 23, 2015 at 21:36 | answer | added | Chris Gerig | timeline score: 5 | |
| Oct 23, 2015 at 19:25 | answer | added | David A. Jackson | timeline score: 2 | |
| Oct 20, 2015 at 10:51 | comment | added | Samarkand | Todd, thank you very much. But I doubt that it is always possible to find such a nice extension. Are there some other tricks? | |
| Oct 20, 2015 at 9:27 | comment | added | Todd Leason | $H=\langle a,[a,b]\rangle$ is normal in $G$ (for, $bab^{-1}=[a,b]^{-1}a$ and $b[a,b]b^{-1}=(bab^{-1})b^2a^{-1}b^{-2}=(bab^{-1})b^2a=(bab^{-1})a^{-1}=[a,b]^{-1}$. Hence there is an extension $$0 \to \mathbb{Z}^2 \to G \to \mathbb{Z}/2 \to 0$$ Therefore the computation of $H_2(G)$ should be within reach. | |
| Oct 19, 2015 at 18:51 | comment | added | Samarkand | For example, how to compute $H_2$ of $G=\langle a,b| [[a,b],a], a^2b^2 \rangle$? | |
| Oct 19, 2015 at 18:50 | comment | added | Samarkand | Of course I know about Hopf's formula. The question is how Hochschild-Serre spectral sequence works, how to define the nature of arrows. | |
| Oct 19, 2015 at 16:55 | comment | added | Todd Leason | @abx: This formular is known as Hopf's Theorem. | |
| Oct 19, 2015 at 8:38 | comment | added | Derek Holt | It is not hard to compute a presentation of a covering group of $G$, which is a central extension of $H_2(G,{\mathbb Z})$ by $G$, but it is not always easy to use that to get a presentation of the central subgroup $H_2(G,{\mathbb Z})$. | |
| Oct 19, 2015 at 7:12 | comment | added | abx | If $F$ is the free group with 2 generators, and $R$ the normal subgroup generated by your relators, the Hochschild-Serre spectral sequence provides an isomorphism $H_2(G,\Bbb{Z})\cong R\cap (F,F)/(F,R)$. A more explicit computation is usually quite difficult, and depends very much on the form of the relators. | |
| Oct 19, 2015 at 6:34 | history | asked | Samarkand | CC BY-SA 3.0 |