Timeline for Is every paracompact, Hausdorff, locally contractible space homotopy equivalent to a CW complex?
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| May 24, 2014 at 1:26 | comment | added | Eric Wofsey | @YonatanHarpaz: I haven't checked completely, but I think you're right. If I'm not mistaken, a small ball around a typical point in $X-U$ will look like a compactification of equatorial strips of some fixed codimension in all of the spheres, and these strips are themselves equivalent to spheres of lower dimension. The only reason this manages to be "locally contractible" in Borsuk's sense is that if you take a slightly larger ball, the codimension of the strips will decrease and so the strips you had before will be contractible inside the new larger strips. | |
| May 23, 2014 at 18:24 | comment | added | Yonatan Harpaz | I think I managed to convince myself that Borsuk's example is not locally contractible in the strong sense. The space X contains a dense open subset $U$ which is homotopy equivalent to a wedge of spheres. Observing the details of the construction, it seems that a generic point in $X - U$ will have neighborhoods which are equivalent to compactified wedges of spheres, and hence not contractible. | |
| May 23, 2014 at 17:22 | comment | added | Jeremy Rickard | Ah, OK. I haven't tried to understand Borsuk's space, but it looks as though @EricWofsey has. Maybe he can say if it's locally contractible in the stronger sense? | |
| May 23, 2014 at 14:28 | comment | added | Yonatan Harpaz | Thanks, although there is still a slight problem with this example. It seems that Borsuk is using a somewhat different definition of locally contractible than what I meant. The definition of locally contractible I had in mind was that every neighborhood $U$ of a point $x$ contains a smaller neighborhood $V$ which is itself contractible. Borsuk is using a similar definition but only requiring that the map $V \to U$ will be null-homotopic. Looking at his example, I cannot tell immediately if it is locally contractible in the stronger sense as well. | |
| May 23, 2014 at 12:24 | comment | added | Eric Wofsey | Borsuk's construction can be found as Theorem 1 here. The idea is to string together spheres of increasing dimension and then compactify that by a Hilbert cube in an appropriate way such that the space remains locally contractible even at the points at infinity. This works because sufficiently small neighborhoods of any point at infinity will not fully contain any of the spheres. | |
| May 23, 2014 at 11:58 | history | answered | Jeremy Rickard | CC BY-SA 3.0 |