Timeline for What space classifies bundles of K(pi,1)'s?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 31, 2013 at 19:40 | vote | accept | John Pardon | ||
| Feb 15, 2013 at 22:57 | comment | added | Ricardo Andrade | @Ulrich: The first example on the nlab page on crossed modules (ncatlab.org/nlab/show/crossed%2Bmodule) might indicate they coincide, at least to someone more knowledgeable of crossed modules than me. | |
| Feb 15, 2013 at 22:43 | comment | added | Ricardo Andrade | I just want to add that the article "Moore-Postnikov towers for fibrations in which $\pi_1$(fiber) is non-abelian" by Richard Hill (available at projecteuclid.org/…) goes into great detail regarding the information in Tom's comment above. It describes the homotopy type of the classifying fibration $BG\to B\textrm{Aut}_*(BG)\to B\textrm{Aut}(BG)$ very thoroughly. | |
| Feb 15, 2013 at 11:03 | answer | added | Ronnie Brown | timeline score: 1 | |
| Feb 15, 2013 at 8:09 | comment | added | Ulrich Pennig | @Tom: How is that related to the classifying space of the $2$-group associated to the crossed module $G \to Aut(G)$? | |
| Feb 15, 2013 at 4:33 | comment | added | Ben Wieland | A lemma I find helpful: if you preserve basepoints, $Aut(B\pi,\*)=Aut(\pi)$ for general groups; $BAut(\pi)$ classifies bundles−with−section. The fiber sequence $Aut(B\pi,\*)\to Aut(B\pi)\to B\pi$ shows that $\pi_0(Aut(B\pi))=Out(\pi)$ and $\pi_1(Aut(B\pi))=Z(\pi)$. | |
| Feb 15, 2013 at 3:38 | answer | added | Peter May | timeline score: 8 | |
| Feb 15, 2013 at 1:33 | comment | added | Tom Goodwillie | Yes, it's just as you think. You can apply the usual bar construction to make a simplicial space from that topological monoid, and get the space you want from that. By the way, $\pi_1$ of this space, which is $\pi_0$ of the monoid, is the group of outer automorphisms of $G$, cokernel of the map $G\to Aut(G)$ that takes $g$ to $x\mapsto gxg^{-1}$. And $\pi_2$ of the space, which is $\pi_1$ of the monoid, is the center of $G$, the kernel of the same map $G\to Aut(G)$. And the rest of the homotopy groups are trivial. | |
| Feb 15, 2013 at 1:27 | comment | added | Misha | How about $K(Out(G),1)$? The (almost) example where this is frequently used is when $G$ is the fundamental group of a closed hyperbolic surface $S_g$, so one uses the moduli space $M_g$ of $S_g$ (regarded as an orbifold) for classification of bundles with surface fiber. Then the orbifold $M_g$ is essentially $K(Map_g, 1)$, where $Map_g$ is the mapping class group of $S_g$, i.e., $Map_g=Out(G)$. | |
| Feb 15, 2013 at 1:08 | history | asked | John Pardon | CC BY-SA 3.0 |