Timeline for How does all of the bundles over a certain manifold characterize the homotopy class of the base manifold?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 22, 2011 at 11:06 | vote | accept | Honglu | ||
| Apr 22, 2011 at 11:06 | vote | accept | Honglu | ||
| Apr 22, 2011 at 11:06 | |||||
| Apr 21, 2011 at 9:14 | comment | added | Honglu | I'm not able to judge. But if it is said to be right, I will accept this. I hope after finishing Hatcher and some references I would be able to read it. | |
| Apr 21, 2011 at 7:36 | history | edited | Oscar Randal-Williams | CC BY-SA 3.0 |
added 843 characters in body
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| Apr 21, 2011 at 3:37 | comment | added | Dylan Wilson | You'd need the space to be simple (i.e. the action of $\pi_1$ to be trivial on higher homotopy groups) in order to apply the homology version of Whitehead here. | |
| Apr 21, 2011 at 2:38 | comment | added | Tom Goodwillie | Wait a minute: The relative Hurewicz Theorem is not that strong. A map inducing isomorphisms on $\pi_1$ and $H_n$ for all $n$ need not be a weak equivalence. | |
| Apr 21, 2011 at 1:47 | comment | added | Honglu | Why does a homology equivalence plus a $\pi_1$ lead to weak homotopy equivalence? | |
| Apr 20, 2011 at 14:35 | comment | added | Tom Goodwillie | You mean, because the loopspace is not quite a group? but there are ways of rectifying that, for example Kan's Quillen adjunction between simplicial groups and reduced simplicial sets. | |
| Apr 20, 2011 at 14:09 | comment | added | Oscar Randal-Williams | They won't typically be equivalent to classifying spaces of actual groups, though. | |
| Apr 20, 2011 at 13:55 | comment | added | Tom Goodwillie | Or instead of using topological groups that are either abelian or discrete, you can use more general ones and say that $M$ and $N$ themselves are equivalent to classifying spaces, and do it that way. This is if they are connected. If they are not, then it is false. | |
| Apr 20, 2011 at 13:48 | history | answered | Oscar Randal-Williams | CC BY-SA 3.0 |