Timeline for A graph manifold without an orientation reversing involution?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 8, 2021 at 10:19 | vote | accept | aglearner | ||
| Nov 7, 2021 at 23:16 | answer | added | jonathan | timeline score: 6 | |
| Nov 4, 2021 at 15:39 | comment | added | Ryan Budney | Orientation-reversing involutions can be subtle. There are lens spaces that do not admit them. For the lens space denoted $L_{p,q}$ where $p$ is the order of the fundamental group, usually the condition is written $q^2 \neq -1 \ mod \ p$. | |
| Nov 4, 2021 at 9:41 | comment | added | aglearner | Thanks @RyanBudney , I started with the most trivial case - $S^1$-bundles over surfaces, and they all have such an involution. Following your suggestion, I've just tried with lens spaces, but don't see why they will give an example. Indeed, a lens space is a quotient $S^3/\mathbb Z_n$ where $\mathbb Z_n$ is acting linearly on $\mathbb C^2$. But it should be possible to find a $\mathbb Z_2$ extension of this action that would also reverse orientation. So your suggestion for lens spaces doesn't seem to work (am I correct?)? If you have a simple example that works, I would be grateful to see it. | |
| Nov 2, 2021 at 2:20 | comment | added | Ryan Budney | There's plenty. You can answer this question completely with a little leg work. The start of the inductive process is to answer it for Seifert-fibered spaces. So why not start with something simple like lens spaces? Automorphisms of lens spaces preserve their genus $1$ Heegaard splitting, up to isotopy. | |
| Nov 1, 2021 at 22:58 | history | asked | aglearner | CC BY-SA 4.0 |