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Given a discrete group $G$ (not assumed to be abelian), is there a nice construction of a topological space which classifies bundles of $K(G,1)$'s? I guess I should take something like $B\operatorname{Aut}(K(\pi,1))$ where $\operatorname{Aut}(K(\pi,1))$ is the monoid of self homotopy equivalences of some fixed $K(\pi,1)$. I don't know if this makes any sense, though.

Morally speaking, things are defined as follows. A bundle of $K(\pi,1)$'s over a space $X$ is a fibration $Y\to X$ whose fibers are $K(\pi,1)$'s. Two bundles of $K(\pi,1)$'s $Y_1,Y_2\to X$ are said to be isomorphic iff there is a map $Y_1\to Y_2$ of spaces over $X$ which is a homotopy equivalence on each fiber.

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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., $Map_g=Out(G)$. –  Misha Feb 15 '13 at 1:27
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. –  Tom Goodwillie Feb 15 '13 at 1:33
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 $\pi_1(Aut(B\pi))=Z(\pi)$. –  Ben Wieland Feb 15 '13 at 4:33
@Tom: How is that related to the classifying space of the $2$-group associated to the crossed module $G \to Aut(G)$? –  Ulrich Pennig Feb 15 '13 at 8:09
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. –  Ricardo Andrade Feb 15 '13 at 22:43
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2 Answers 2

up vote 7 down vote accepted

For any space $F$ you can form the topological monoid $Aut(F)$ and take its classifying space. That will classify Hurewicz fibrations with fiber $F$. A little care is needed since $Aut(F)$ won't generally have the homotopy type of a CW complex unless $F$ is a finite CW complex, which $K(G,1)$ is usually not. For the proof it is sensible to take base spaces of the homotopy types of CW complexes. One source for a complete proof is "Classifying spaces and fibrations", #15 on my web page. (There are two obvious notions of the right equivalence relation on fibrations, but these give the same answer, as shown in "Fiberwise localization and completion'', #31 on my web page.)

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To put another context to the above comments and answers, recall that a crossed module consists of a morphism $\mu: M \to P$ together with an action of $P$ on the right of the group $M$ satisfying the two axioms

  1. $\mu(m^p)= p^{-1}\mu(m) p$;

  2. $ m^{-1} nm = m^{\mu n}$,

for all $m,n \in M, p \in P$. This definition is due to J.H.C. Whitehead in 1946.

Examples of crossed modules are:

  1. the inner automorphism crossed module $\chi: M \to Aut(M)$ for any group $M$;

  2. the inclusion $M \to P$ of a normal subgroup $M$ of $P$;

  3. the zero map $0: M \to P$ for any right $P$-module $M$;

  4. the induced map $\pi_1(F) \to \pi_1(E)$ for any pointed fibration $F \to E \to B$;

and others!

Any such crossed module has a classifying space $B(M \to P)$ whose homotopy groups are trivial above dimension $2$, and with $\pi_1 \cong Coker \mu$, $\pi_2 \cong Ker \mu$. More details of these facts are in the book EMS Tract 15. Crossed modules are equivalent to group objects in the category of groupoids, and this gives one way of defining the classifying space, using bisimplicial sets, and are conveniently regarded as $2$-dimensional versions of groups, since they model pointed weak homotopy $2$-types. Note that the second homotopy group, even considered as a module over $\pi_1$, is generally but a pale shadow of the homotopy $2$-type.

There is also a Seifert-van Kampen type theorem with values in crossed modules, and this allows some computations of nonabelian second relative homotopy groups. See again the EMS Tract 15.

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Hello! I'm reading your book Nonabelian algebraic topology and encountered the notion of a crossed module over a groupoid in section 6.2, where the first condition is that $$\mu(x^a)=\mu(x)^a$$ However, the action of $a$ is on elements in $M$. Am I right in assuming that this is just $$\mu(x^a)=-a+\mu(x)+a$$ similar to the case of groups ? –  Stefan Hamcke Jul 18 at 11:20
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