Timeline for Why is any $G$-resolution a principal $G$-bundle?
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| Feb 25, 2018 at 20:42 | comment | added | Jrnm | As @MarkGrant pointed out the free condition is indeed the one that guarantees local triviality. The key point is to notice that every orbit of a point in $X_{n} - X_{n-1}$ has a unique representative in $D_{n} - X_{n-1}$. This way the trivializing cover would be the open sets of the form $p(X_{n} - X_{n-1})$. In my opinion the definition of a $G$-resolution becomes clearer when thinking of Milgram's classifying spaces. It also would be nice to know where one could find the mimeographed notes | |
| Feb 23, 2018 at 22:26 | history | edited | Jrnm |
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| Feb 23, 2018 at 22:11 | comment | added | Mark Grant | I can't find a reference (there are some mimeographed notes by Rothenberg and Steenrod of the same name, but they don't seem to be online) so I'll just leave a comment. The definition of $G$-resolution seems to be modeled on Milnor's construction of the classifying space using joins. The proof they had in mind is probably a generalisation of Milnor's proof in "Constructions of Universal Bundles, II". In particular, rather than using 2. to show that the action is free and topologically nice, try using 2. to show that the projection is locally trivial. | |
| Feb 23, 2018 at 21:13 | history | edited | Ben McKay | CC BY-SA 3.0 |
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| Feb 23, 2018 at 20:15 | history | asked | Jrnm | CC BY-SA 3.0 |