Timeline for Homeomorphisms of $S^n\times S^1$
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 17, 2021 at 7:49 | comment | added | Ryan Budney | Small comment: the "up to pseudo-isotopy" part of your question is essential. If you replace it with pure isotopy, then the answer to the question would be no. | |
| Feb 5, 2017 at 12:30 | comment | added | olga | I corrected the question | |
| Feb 5, 2017 at 11:17 | comment | added | Dave Witte Morris | Then I think there is a mistake in the revised statement of your question: it says you want there to exist at least one sphere that is preserved, not all spheres, because it says "for some $x$", not "for all $x$". Please fix this if you really do mean "all" instead of "some". (Also, I don't see where you use the assumption that $h$ preserves orientation, because the homeomorphism that reverses the orientation on $S^n$ preserves all spheres.) | |
| Feb 5, 2017 at 10:50 | comment | added | olga | I need that h is pseudo isotopic to a homeomorphism g which preserves all spheres. And it is true when h preserves orientation and acts identity in $\pi_1$. But, unfortunately, all results, ecxept n=3, are true not for homeomorphisms but for PL-homeomorphisms. Now I need some result in the direction that there is no difference between these groups. | |
| Feb 4, 2017 at 22:43 | comment | added | Dave Witte Morris | Isn't it true that all three standard generators fix $S^n \times \{0\}$? If so, then doesn't your argument show that there is no need for any assumptions on $h$? (And your argument also seems to show that if $h$ acts trivially on $\pi_1$, then it is possible to get the condition on $g$ for all $x$, not just some $x$, without assuming that $h$ is orientation preserving.) | |
| Feb 4, 2017 at 6:09 | comment | added | olga | With my pleasure. From pages 71-72 follows that the group of homeomorphisms under pseudo isotopy is isomorphic to $Z_2\times Z_2\times Z_2$. As I suppose that my homeomorphism preserve orientation in act identity in $\pi_1$ that their group is isomorphic to $Z_2$ and we now stamdart homeomorphism in each class: identity and with rotation (here arxiv.org/pdf/0903.1488v2.pdf it is possible to see some explanation for n=2 on page 2), both preserve spheres $S^n$. | |
| Feb 3, 2017 at 13:47 | comment | added | Igor Belegradek | Could you explain how what is written on pp.71-72 answers your question? The paper is at maths.ed.ac.uk/~aar/papers/kervsurv.pdf. | |
| Feb 3, 2017 at 5:34 | history | answered | olga | CC BY-SA 3.0 |