Timeline for How can I visualize the nontrivial element of $\pi_4(S^3)$ and $\pi_5(S^3)$ ?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Sep 21, 2010 at 6:56 | comment | added | Makoto | It is not fair to revise my post after someone noticed it, though I found some difficulty with "complexification". | |
| Sep 21, 2010 at 5:59 | comment | added | S. Carnahan♦ | You might want to revise the first sentence, since $S^3$ is isomorphic to $SU(2)$, not the quotient $SO(3)$. | |
| Sep 20, 2010 at 21:25 | comment | added | Makoto | If you mean by "visualize" some graphs like string topology and higher category, you can see many pictures of Riemann surfaces, trees (operad), and braid groups. But I think it is not the ordinary mathematics based on logical algebra. Also, torsion part is not so difficult to handle and I didn't care about it. In my opinion, the "theoretical tool" is the jet space. | |
| Sep 20, 2010 at 6:01 | comment | added | Makoto | I thought Yuji wanted several other comments, but if he wanted geometric topology, I should mention lens space, which I am not working on. (Should I call it by Spin group?) I was considering whether I should write about loop group and DGA model, but it is outside of my reach today. (And I am not working on differential topology of exotic space today.) | |
| Sep 20, 2010 at 5:44 | comment | added | Somnath Basu | I'm finding it hard to parse your answer! For instance, $SO(3)$ has $SU(2)$ as its double cover and not the other way around. What higher dimensional terms are you referring to and how does Bott periodicity come in? I can imagine referring to Bott's result about the cell structure of $\Omega G$ but that's not clear either. And nothing that you have said helps to visualize the higher homotopy elements. One, of course, knows that there are algebraic tools to compute what the referred elements are but that wasn't the question! | |
| Sep 20, 2010 at 5:40 | history | edited | Makoto | CC BY-SA 2.5 |
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| Sep 20, 2010 at 5:29 | history | answered | Makoto | CC BY-SA 2.5 |