Timeline for Aspherical manifold with superperfect fundamental group and non-trivial center?
Current License: CC BY-SA 4.0
21 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 30, 2019 at 10:29 | answer | added | Sam Nead | timeline score: 3 | |
| Dec 30, 2019 at 5:07 | comment | added | Lee Mosher | Someone should turn this discussion into an answer. | |
| Dec 30, 2019 at 3:18 | history | edited | Jeffrey Rolland | CC BY-SA 4.0 |
[Edit removed during grace period]
|
| Dec 20, 2019 at 22:45 | history | edited | Jeffrey Rolland | CC BY-SA 4.0 |
added 4 characters in body
|
| Dec 20, 2019 at 22:39 | comment | added | Jeffrey Rolland | @YCor Edited into OP | |
| Dec 20, 2019 at 22:38 | history | edited | Jeffrey Rolland | CC BY-SA 4.0 |
added 122 characters in body
|
| Dec 20, 2019 at 22:09 | comment | added | Jeffrey Rolland | @DannyRuberman Wow, that's awesome, thank you! I'm actually going to take a fiber bundle with a torus as the base manifold and need the total space to be at least 6-dim'l at some point, so I could just use a 3-dim'l torus or a 2-dim'l torus and a higher-dim'l example of what you just suggested; it's kind-of "six of one, half a dozen of another" to me at this point. However, modulo the dimension of the manifold, this is exactly what I needed, so thank you | |
| Dec 20, 2019 at 19:00 | comment | added | Danny Ruberman | See section 2.3 of the Boileau-Zieschang paper; you'll need to compute the Seifert invariants first. You can find the answer in section 3.5.1 of Saveliev's book, Invariants for homology 3-spheres. Did you only care about the 3-dimensional case? My comment above gives examples for dim = 0 mod 3, but probably there are ways to do dim = 1 or 2 mod 3. | |
| Dec 20, 2019 at 17:38 | comment | added | Jeffrey Rolland | @DannyRuberman Just a finite presentation of the fundamental groups; I'll need that for the construction I'm doing | |
| Dec 20, 2019 at 16:06 | comment | added | Danny Ruberman | I think SnapPy is all about hyperbolic manifolds. What are you hoping to learn/compute about those examples using SnapPy? | |
| Dec 20, 2019 at 16:02 | comment | added | Danny Ruberman | One more point--there are well-known genus 2 Heegaard splittings (derived from well-known surgery diagrams) for the Brieskorn homology spheres with 3 exceptional fibers. See eg Boileau-Zieschang [link] (numdam.org/article/AST_1988__163-164__247_0.pdf) | |
| Dec 20, 2019 at 16:00 | comment | added | Jeffrey Rolland | Thanks so much for the responses. If I may "go to the well" one more time, how may I specify a Brieskorn homology sphere in SnapPy? | |
| Dec 20, 2019 at 15:54 | comment | added | HJRW | @DannyRuberman — of course. I said that $\mathbb{Z}$ is “the best you can do”, because the OP was hoping for $\mathbb{Z}/2$. | |
| Dec 20, 2019 at 15:52 | comment | added | Danny Ruberman | @HJRW Perhaps you mean that the center (if non-trivial) must contain $\mathbb{Z}$? Since the OP didn't specify the dimension one could take products of Seifert fibered spaces. This produces examples in dimension $3n$ with center $\mathbb{Z}^n$. | |
| Dec 20, 2019 at 15:46 | comment | added | Jeffrey Rolland | @HJRW Thanks so much for the Brieskorn homology spheres examples! I really appreciate it! | |
| Dec 20, 2019 at 15:44 | comment | added | Jeffrey Rolland | @HJRW That's right, and the hyperbolic ones, at least (I think?), can't have a $\mathbb{Z} \times \mathbb{Z}$ or higher rank free Abelian group as a subgroup; I had looked up both those facts last night but forgot them. It's truly $\mathbb{Z}$ or nothing, thanks. | |
| Dec 20, 2019 at 15:38 | comment | added | HJRW | The fundamental groups of aspherical manifolds are always torsion free for homological reasons, so a centre of $\mathbb{Z}$ is the best you can do. | |
| Dec 20, 2019 at 15:32 | comment | added | YCor | Glossary: $G$ superperfect means $H_1(G,\mathbf{Z})=H_2(G,\mathbf{Z})=0$. | |
| Dec 20, 2019 at 15:32 | comment | added | HJRW | In every dimension, the fundamental group of a hyperbolic manifold has trivial centre. There are no examples of what you want in dimension 2, but in dimension 3, there are the Brieskorn homology spheres $\Sigma(p,q,r)$ for $1/p+1/q+1/r <1$. These are aspherical homology spheres which are Seifert fibred, so their fundamental groups have centre $\mathbb{Z}$. | |
| Dec 20, 2019 at 15:32 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title, fixed typos
|
| Dec 20, 2019 at 15:17 | history | asked | Jeffrey Rolland | CC BY-SA 4.0 |