Timeline for Contractible set in a manifold
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 25, 2021 at 1:55 | vote | accept | Zhiqiang | ||
| Jul 16, 2021 at 13:04 | comment | added | Tom Goodwillie | I believe it is true in the case n=2. The argument that I have in mind uses the fact that an embedded circle in a closed surface cannot be nullhomotopic if it separates the surface into two pieces neither of which is a disk. | |
| Jul 15, 2021 at 21:49 | comment | added | Shijie Gu | I just want to add more ideas to n = 2 case. Once we had the nbhd $U$ of $K$ as I described above. We may lift it to $\tilde{U}$ in the universal cover $\tilde{M}$. Then we can work on $\tilde{M}$, essentially $R^2$. Build a compact 2-manifold with boundary $H$ in $\tilde{M}$ such that $K \subset Int H \subset H \subset \tilde{U}$. If $\partial H$ is connected, by Schonflies theorem, $H$ is a 2-cell. If $\partial H$ is not connected, one may cut away at $H$ until it is connected. | |
| Jul 15, 2021 at 21:25 | comment | added | Shijie Gu | @TomGoodwillie: Ah, I see. Sorry. My mistake. I'll delete the comment. | |
| Jul 15, 2021 at 20:00 | comment | added | Tom Goodwillie | Oh, I see. You're just saying that if K is contractible in M then some neighborhood of K is contractible in M. But then are you also using this remark (which is true for every n) to answer the question when n=2? | |
| Jul 15, 2021 at 17:55 | comment | added | Shijie Gu | @TomGoodwillie: Define a map $f: A \to M$ on $A = (K \times [0,1]) \cup (M \times \{0,1\})$ of $M \times [0,1]$ as $f(<m,0>) = m$ and $f(<m,1>) = F_1(K) = q$ for $m\in M$ and as $f(<k,t>) = F_t(k)$ for $k \in K$. Since $M$ is an ANR, $f$ extends to a map $F': V \to M$ defined on a nbhd $V$ of $A$ in $M \times [0,1]$. Then $K$ has a nbhd $U$ with $U \times [0,1]$ in $V$, and the restriction on $U\times [0,1]$ is the desired contraction. | |
| Jul 15, 2021 at 16:27 | comment | added | Tom Goodwillie | @Shijie Gu: I think you misunderstood the hypothesis. $K$ is not contractible, but "contractible in $M$". | |
| Jul 14, 2021 at 2:29 | comment | added | Zhiqiang | Nice example! Is the original conclusion correct if the dimension $n=2$? | |
| Jul 13, 2021 at 15:20 | history | answered | Danny Ruberman | CC BY-SA 4.0 |