Timeline for Homotopy equivalence vs weak homotopy equivalence in Gromov's h-principle
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 31, 2014 at 5:45 | comment | added | Marc Nardmann | That is a useful reference, thank you. It mentions also several necessary conditions for $C(X,Y)$ to be of CW type, for instance Example 3 on p. 25. Let us take a trivial bundle $E=M\times F\to M$ and, say, $\mathcal{R}=J^kE$. Then $Sec(\mathcal{R})$ is homotopy equivalent to $C(M,F)$. Suitable manifolds $M,F$ are up to homotopy equivalence of the form of Example 3, e.g. with $T=\mathbb{R}$, all $m_\lambda=1$, $dim M=2$ and $Y=S^2$. Then $C(M,F)$ is not of CW type. I.e., the Eliashberg/Mishachev argument fails in general. Of course, the inclusion might still always be a homotopy equivalence. | |
| Jul 30, 2014 at 14:43 | comment | added | Igor Khavkine | It was just a thought, so you may be right. It seems that the CW type of mapping spaces is an old and well studied question. In particular, it seems that $C(X,Y)$ is of CW type when both $X$ and $Y$ are of CW type, with $X$ finite dimensional and $Y$ has only finitely many non-vanishing homotopy groups (not sure how to check that last condition, unfortunately): arxiv.org/abs/0708.2838 | |
| Jul 30, 2014 at 0:00 | comment | added | Marc Nardmann | Hi Igor, I see a priori no reason why countable intersections of "good" spaces in our context should be good (where being good means at least that weak homotopy equivalence implies homotopy equivalence). I would rather suspect that in general they are not (whatever "good" means precisely). Of course, even if they are not good, that will not tell us whether our inclusions (which are quite special weak homotopy equivalences) are always homotopy equivalences. | |
| Jul 29, 2014 at 3:49 | comment | added | Igor Khavkine | If $M$ is compact, then $Sec(\mathcal{R})$ is an open subset of $Sec(\mathcal{J^kE})$, hence a manifold with the compact-open topology. Of course, the same cannot be said when $M$ is non-compact, unless the fibers of $\mathbb{R}$ differ from those of $J^kE$ only over a compact set $K\subset M$. On the other hand, we can write $Sec(\mathcal{R}) = \bigcap_K Sec(\mathcal{R}_K)$, where $K$ ranges over a countable exhaustion of $M$ by compacts, if one exists, and $\mathcal{R}_K$ agrees with $\mathcal{R}$ over $K$ and with $J^kE$ outside it. Could that be enough for a CW structure? | |
| Jul 27, 2014 at 22:48 | history | asked | Marc Nardmann | CC BY-SA 3.0 |