Timeline for Is this generalization of Borsuk Ulam true? Roots of unity
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 19, 2017 at 10:00 | comment | added | Ali Taghavi | @NeilStrickland So my apology for not reading your answer, carefully.I read it again.I am on my phon. | |
| Oct 19, 2017 at 7:06 | comment | added | Neil Strickland | @AliTaghavi There are no even spheres involved here at all. There is a free action of $C_3$ on $SO(3)$ (which is almost the same as $S^3$) and a free action of $C_3$ on $\mathbb{C}^\times$ (which is homotopy equivalent to $S^1$). There is no issue about the existence of these actions, the point is to prove that there is no $C_3$-equivariant map $SO(3)\to \mathbb{C}^\times$. | |
| Oct 19, 2017 at 6:47 | comment | added | Ali Taghavi | Are you proving that there is no a free action of $\mathbb{Z}/3\mathbb{Z}$. If yes how this solve the OP question? Moreover I think there is an easy argument for non existence of such free action, as I learn in the book by Allen Hatcher: n=2 is the only $n$ such that $\mathbb{Z}/n\mathbb{Z}$ can act freely on a even domensional sphere. The proof is simply based on the degree of maps. The "degree" is a group isomirphism from $G$ to$ {1,-1}$ (Counting $g\in G$ as a homeomorphism of even sphere. On the other hand a fixed point free homeomorphism has degree -1. | |
| Oct 18, 2017 at 20:19 | comment | added | Andy | Hi, I am very sad to say I don't know enough about topology to understand your answer, but it motivates very much to learn about it. (I accepted your answer assuming that upvotes from other people verify your solution). | |
| Oct 18, 2017 at 20:16 | vote | accept | Andy | ||
| Oct 18, 2017 at 17:27 | history | answered | Neil Strickland | CC BY-SA 3.0 |