Timeline for index of morse functions and homotopical dimension
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Oct 8, 2010 at 21:01 | comment | added | Ryan Budney | Mazur manifolds (at least, ones that can be represented as $S^1 \times D^3$ union a $2$-handle) are all smooth submanifolds of $S^4$. On the Wikipedia page the reason is sketched in the section titled "Mazur's Observation". When you refer to configuration spaces, are you trying to go after a Salvatore-Longoni type problem, looking at the homotopy-type of configuration spaces of a Mazur manifold, that kind of thing? | |
| Oct 8, 2010 at 16:44 | comment | added | Victor | Thank you Ryan. As usual you give a very interesting answer. In relation with that, don't you know what is known about the configuration spaces of these manifolds? And what about their embeddings into $R^3$, $R^4$, $R^5$...? | |
| Oct 7, 2010 at 16:28 | comment | added | Ryan Budney | In full generality there's likely a natural Whitehead-torsion obstruction to doing what you want to do, given by the s-cobordism theorem. That would require more thought than I'm prepared for this morning. | |
| Oct 7, 2010 at 16:12 | comment | added | Ryan Budney | I believe it's unknown whether or not there are exotic smooth structures on open 4-balls that extend to smooth structures on a compact 4-ball so that would be a potential counter-example for $k=0$. Less hypothetically, Mazur manifolds are counter-examples. In high dimensions if your manifold and its boundary are simply-connected the minimal handle presentation theorem would kick in and say there are no counter-examples. But once you have $\pi_1$ in either the manifold or the boundary there are potential obstructions. I wrote up a little thing on Mazur manifolds on Wikipedia. | |
| Oct 7, 2010 at 14:40 | comment | added | Victor | Can one find a counterexample among interiors of compact manifolds with boundary? | |
| Oct 5, 2010 at 18:59 | vote | accept | Victor | ||
| Oct 5, 2010 at 18:15 | history | edited | Ryan Budney | CC BY-SA 2.5 |
more detailed notes. ; added 106 characters in body
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| Oct 5, 2010 at 17:45 | history | edited | Ryan Budney | CC BY-SA 2.5 |
added 130 characters in body
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| Oct 5, 2010 at 17:37 | history | answered | Ryan Budney | CC BY-SA 2.5 |