Timeline for Approximate classifying space by boundaryless manifolds?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Apr 1 at 19:46 | vote | accept | 0207 | ||
| Apr 1 at 18:59 | history | became hot network question | |||
| Apr 1 at 17:54 | history | edited | LSpice | CC BY-SA 4.0 |
Link to comment; typos
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| Apr 1 at 17:48 | answer | added | mme | timeline score: 11 | |
| Apr 1 at 16:56 | comment | added | 0207 | @mme Yes we can adopt this definition. | |
| Apr 1 at 16:50 | comment | added | mme | Here is a formal statement which I think the OP would be happy with as a definition of "approximate". For any compact Lie group $G$ and any integer $n$, is there some closed manifold (of large dimension) $M$ which is $n$-connected and carries a free action of $G$? | |
| Apr 1 at 16:47 | comment | added | Moishe Kohan | What exactly do you mean by "approximate?" | |
| Apr 1 at 16:41 | history | edited | 0207 | CC BY-SA 4.0 |
added 326 characters in body; edited title
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| Apr 1 at 16:35 | comment | added | 0207 | @Achim Krause Thanks, this is an excellent answer. I just realized I'm more interested in the converse problem. Given a Lie group $G$ and its classfying space $BG$, can we always approximate it using boundaryless manifolds? | |
| Apr 1 at 12:49 | comment | added | Ryan Budney | Which definition of $BG$ are you using to get a well-defined homeomorphism type? Usually $BG$ is only well-defined up to a homotopy-equivalence. So $BG = \{*\}$ and $BG=[0,1]$ are often seen as basically "the same". | |
| Apr 1 at 4:25 | comment | added | Achim Krause | But I think $BG$ is basically never finite dimensional. For example for compact connected $G$, this is a nice exercise with the Serre spectral sequence. | |
| Apr 1 at 4:23 | comment | added | Achim Krause | Any finite CW complex is homotopy equivalent to a manifold with boundary (by embedding into $\mathbb{R}^n$ and thickening), and so every finite type CW complex can be approximated by manifolds with boundary. | |
| S Apr 1 at 3:44 | review | First questions | |||
| Apr 1 at 6:41 | |||||
| S Apr 1 at 3:44 | history | asked | 0207 | CC BY-SA 4.0 |