Timeline for Compact metrizable contractible locally contractible topological space of finite covering dimension is a CW complex
Current License: CC BY-SA 4.0
7 events
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| Aug 18, 2020 at 13:10 | comment | added | Igor Belegradek | Yes, contractible manifolds are smoothable and hence CW, but the method of proving this is manifold specific (smoothing theory) and won't help you in the setting of finte-dimensional ANRs, which is the point I was trying to make. | |
| Aug 18, 2020 at 6:30 | comment | added | user145520 | @IgorBelegradek I don't think it's an open problem whether a contractible manifold is homeomorphic to a CW complex. | |
| Aug 18, 2020 at 3:58 | comment | added | HenrikRüping | also the compactification of the tree of which $F_2$ acts, is a counterexample. | |
| Aug 18, 2020 at 1:38 | comment | added | Igor Belegradek | To show that this example has no CW structure note that the space is compact, so the complex is finite, which easily leads to a contradiction. | |
| Aug 18, 2020 at 0:09 | comment | added | Igor Belegradek | It seems the following is a counterexample: Let $X$ be the union of the straight line segments $I_n$ in $\mathbb R^2$ where $I_n$ joins the origin and the point $(1, 1/n)$, where $n$ is a nonnegative integer, and $I_0$ joins the origin and $(1,0)$. | |
| Aug 17, 2020 at 23:56 | comment | added | Igor Belegradek | As you probably know, it is an open problem whether a closed topological $4$-manifold is homeomorphic to a CW complex. I gather you are asking for a countexample in a larger class of spaces. Your assumptions imply that $X$ is an ANR.. | |
| Aug 17, 2020 at 22:49 | history | asked | user145520 | CC BY-SA 4.0 |