Timeline for Null-homotopicness of an inclusion map
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Mar 24, 2023 at 18:06 | comment | added | Power of Topology | Thanks a lot. I will thick that proof out. | |
| Mar 24, 2023 at 16:40 | comment | added | Igor Belegradek | I think, you can consider the simplcial analog of mapping cylinder of the homotopically nontrivial map $S^6\to S^4$. See e.g. Hatcher's "Algebraic topology", p.183 (the proof of theorem 2C.5 on simplicial approximation of CW complexes). It will be homotopy equivalent to $S^4$ and will contain a simplicial complex PL homeomorphic to $S^6$, and I think it will have dimension $7$ (check this). | |
| Mar 24, 2023 at 15:22 | comment | added | Power of Topology | Thanks! This is very helpful. Sorry about being unclear. Basically, I was asking for a sufficient condition that the inclusion map from $L$ to $ K$ is null-homotopic given $L\simeq S^6$ and $K\simeq S^4$. Or any conjecture? I hope this makes sense. | |
| Mar 24, 2023 at 15:12 | comment | added | Igor Belegradek | I am not sure I understand the question; my quess is that you are asking for a minimal $n$ such that any continuous map $S^6\to S^4\times D^n$ is homotopic to a PL embedding. I don't know the minimal one, but $n=6$ is enough according to Irwin's "Embeddings of polyhedral manifolds" jstor.org/stable/1970560. For $n\ge 9$ the result is true by general position. | |
| Mar 24, 2023 at 14:56 | comment | added | Power of Topology | Thank you for the answer. Is there any result on the minimal $n$ such that the inclusion map from $S^6$ to $S^4\times D^n$ to be nontrivial? | |
| S Mar 23, 2023 at 23:19 | history | suggested | J. W. Tanner | CC BY-SA 4.0 |
changed spelling in title
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| Mar 23, 2023 at 22:28 | review | Suggested edits | |||
| S Mar 23, 2023 at 23:19 | |||||
| Mar 23, 2023 at 21:32 | review | Close votes | |||
| Apr 1, 2023 at 3:11 | |||||
| Mar 23, 2023 at 14:56 | comment | added | Igor Belegradek | The answer is not necessarily. The point is that $\pi_6(S^4)\cong\mathbb Z_2$ and the homotopically nontrivial map $S^6\to S^4$ can be homotoped to an inclusion after replacing $S^4$ with $S^4\times D^n$ for sufficiently large $n$. | |
| S Mar 23, 2023 at 14:30 | review | First questions | |||
| Mar 23, 2023 at 15:09 | |||||
| S Mar 23, 2023 at 14:30 | history | asked | Power of Topology | CC BY-SA 4.0 |