Timeline for Uniqueness of compactification of an end of a manifold
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 30, 2013 at 15:17 | vote | accept | Igor Khavkine | ||
| Mar 30, 2013 at 15:17 | comment | added | Igor Khavkine | Having occasionally thought about this question some more, I think this answer is quite close to what I was looking for. In particular, if the hypotheses are strengthened such that the two possible boundaries are $s$-cobordant (rather than just $h$-cobordant), then the two boundaries are not just diffeomorphic, but one can be transformed into the other by a sequence of blow-up and collapse simple homotopy moves. en.wikipedia.org/wiki/… | |
| Nov 24, 2011 at 0:01 | comment | added | Igor Khavkine | I see a finer relationship between $\partial\bar{M}_i$ than the mere existence of an $h$-cobordism. If $\mathcal{F}$ is a closed filter in the interior of the $\bar{M}_i$ (identified via $\psi$) that converges to a closed set $F_0$ in $\partial\bar{M}_0$, then it must converge to another closed set $F_1$ in $\partial\bar{M}_1$. Thus, closed sets in $\partial\bar{M}_0$ can be put into correspondence with closed sets in $\partial\bar{M}_1$. How complicated could the correspondence be? Could it be induced by a homeo- or diffeomorphism up to expansion or collapse of some points (simple homotopy?)? | |
| Nov 23, 2011 at 23:50 | comment | added | Igor Khavkine | Thanks! This is very close to the kind of information I was looking for. I guess I should try to understand the properties of $h$-cobordisms better now. It would be ideal for my purposes if a diffeomorphism $\psi$, like in your answer, would induce by continuity a homeo- or diffeomorphism $\partial\bar{M}_0\to\partial\bar{M}_1$, but that's obviously impossible. In fact, I think $\psi$ does not even always extend to a map. But I'd like to see something as close to such an extension of $\psi$ as possible. | |
| Nov 23, 2011 at 17:25 | history | answered | Oscar Randal-Williams | CC BY-SA 3.0 |