Timeline for CW complexes and paracompactness
Current License: CC BY-SA 2.5
14 events
| when toggle format | what | by | license | comment | |
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| Jun 25, 2011 at 9:12 | history | edited | Stephen S |
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| Feb 9, 2011 at 19:27 | vote | accept | Cary | ||
| Nov 5, 2010 at 7:44 | answer | added | Neil Strickland | timeline score: 9 | |
| Nov 5, 2010 at 1:16 | comment | added | David Roberts♦ | (cont..) otherwise if U is not numerable one just get a classifying map in the homotopy category, not in Top: $B\check{C}(U) \to X$ is only a weak homotopy equivalence. | |
| Nov 5, 2010 at 1:14 | comment | added | David Roberts♦ | @David Carchedi - No. As far as I am aware, no one has identified numerable covers as a pretopology up until now. Dold's book is good for results on numerable covers (as is his original paper 'Partitions of unity in the theory of fibrations'). There is a link to bundle classification via anafunctors hidden in Segal's paper on categories and classifying spaces: the result that if $U \to X$ is a numerable cover, then $B\check{C}(U) \to X$ is a shrinkable map, and so from a cocycle $\check{C}(U) \to \mathbf{B}G$ one gets a classifying map $X \to BG$. | |
| Nov 4, 2010 at 23:37 | comment | added | David Carchedi | @David Roberts: Ahhh... this answers some things I have been wondering. You know any reference where they speak about the Grothendieck topology of enumerable covers like this? | |
| Nov 4, 2010 at 22:37 | comment | added | David Roberts♦ | On the other hand, given a G-bundle on a non-paracompact space, it is the pullback of the universal G-bundle if and only if it trivialises over a numerable cover. This is one place where you have to use a genuinely different Grothendieck topology on Top other than the one involving arbitrary open covers. The two coincide for paracompact spaces, so the difference is subtle and waved over by most authors. | |
| Nov 4, 2010 at 22:34 | comment | added | David Roberts♦ | More accurately, the bundle homotopy theorem needs the bundles to be trivalisable over a numerable cover, and since EG -> BG can be constructed as such, any bundle which is (iso to one which is) pulled back along a classifying map X -> BG is also trivialisable over a numerable cover. Thus you have two bundles on X which can be jointly trivialised over the same numerable cover, and by theorems of Dold on so-called stackable covers (see his book on algebraic topology) I think you get the two bundles to be isomorphic. | |
| Nov 4, 2010 at 20:36 | comment | added | Cary | since the usual proof that two bundles are isomorphic uses the bundle homotopy theorem, and this theorem needs paracompactness. | |
| Nov 4, 2010 at 20:35 | comment | added | Cary | In the homotopy category of CW-complexes, every object is actually a CW complex, right? Because then principal G-bundles are represented by maps into BG. If we enlarge the class of objects to spaces with the homotopy type of a CW-complex, then each space X has a homotopy equivalence from a CW complex K, pulling back the bundle gives a bundle over K, and then we classify this with a map $K \rightarrow BG$. Composing maps, we get a "classifying map" $X \rightarrow BG$. The problem, though, is that pulling back the bundle over $BG$ to a bundle over X doesn't necessarily give an isomorphic bundle, | |
| Nov 4, 2010 at 20:20 | answer | added | David Roberts♦ | timeline score: 10 | |
| Nov 4, 2010 at 19:33 | comment | added | David Carchedi | I think this question is fine. Using Brown's representability theorem, we get, for example, that the functor which assigns a space the set of isomorphism classes of prinicipal G-bundles is representable over the homotopy categeory of CW-complexes. But also, we know that the classifying space of G represents this same functor on paracompact Hausdorff spaces. So he is asking what is going on here... | |
| Nov 4, 2010 at 18:56 | comment | added | Ryan Budney | Paracompactness isn't a homotopy invariant. But CW complexes are paracompact, this is a common lemma in most intro algebraic topology texts. Your question is perhaps most appropriate for the math.stackexchange website. | |
| Nov 4, 2010 at 18:46 | history | asked | Cary | CC BY-SA 2.5 |