Timeline for Tubular neighbourhoods are unique up to ambient isotopy?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jan 30, 2022 at 18:09 | comment | added | Ben McKay | @StefanFriedl: Wall's proof of the existence of tubular neighborhoods doesn't work for me. His Theorem 2.3.2 p. 45 uses Corollary A.2.6, which seems incorrect since it doesn't require $X$ closed, and he proves A.2.6 using A.2.5, which assumes $X$ compact. I am not sure that A.2.5 works for closed noncompact $X$, and I think A.2.6 fails for nonclosed $X$. | |
| Nov 11, 2021 at 23:44 | comment | added | horned-sphere | Tubular neighborhoods are unique up to isotopy, but different framings of the tubular neighborhood of the knot give different 2-handle attachments. The post asks about this too, so I just wanted to say I would love a reference for that as well. | |
| Nov 18, 2019 at 12:09 | comment | added | mathquest | @StefanFriedl Thank you, that does seem like a nice textbook. It's the same definition as in Hirsch. Wall, though, also shows that this definition is equivalent to the Riemannian one. | |
| Nov 16, 2019 at 10:05 | comment | added | Stefan Friedl | I think the best treatment of tubular neighborhoods is given in Wall's recent book on differential topology, in particular he gives an intrinsic definition of a tubular neighborhood (linear disk bundle) instead of the very awkward definition via Riemannian metric. See Theorem 2.5.5 of Wall. | |
| Nov 11, 2019 at 14:25 | comment | added | Igor Belegradek | I forgot to add that the isotopy can be made ambient by the isotopy extension lemma (chapter 8 of Hirsch's text). | |
| Nov 11, 2019 at 14:19 | comment | added | mathquest | @IgorBelegradek Thank you! This reference additionally obtains a bundle map at the end of the isotopy. | |
| Nov 11, 2019 at 12:23 | comment | added | Igor Belegradek | Tubular neighborhoods are unique up to isotopy. Search for "tubular neighborhod" in the index of M. Hirsch's "Differential topology". | |
| Nov 11, 2019 at 4:45 | history | asked | mathquest | CC BY-SA 4.0 |