I decided to write down this nice exercise since it is the kind of exercise which is readily solved with the right tools (i.e. l.e.s. of a fibration), but can become non-trivial if one doesn’t know them.
In particular, I want to prove that the Eilenberg MacLance space , for a discrete group is a model for the classifying space .
First of all, using l.e.s. this exercise is trivial: in fact we have the following fibration
gives us the long exact sequence of homotopy groups. Recall that and for ( is discrete). By definition is contractible, therefore the long exact sequences boils down to
This proves that is a model for the .
what happens if we don’t know this? Ok, it’s an easy result everyone should know, but it’s good to have a proof which relies on a totally different approach, since it’s the perfect chance to revise some notions in covering theory.
Our approach will be the following: given , we will show that it has the universal property of the classifying space . Let us start with the following crucial observation: begin discrete, a principal -bundle over is in particular a covering space of . As I said before, this is crucial and it’s a good sign that it is crucial otherwise we would be able to find counterexamples with the above proof. So, this is our situation so far:
we want to find a map which classifies . The first step to build such map is to observe that, as a covering, is classified by and such is automatically a normal covering of . In fact, by definition of principal bundle, we have a fibre-wise transitive -action on , but this is exactly the requirements to be a normal covering (see [Hat] page 70).
Proof: first of all, the quotient we wrote above makes sense since we have shown a moment ago that was a normal covering of . This means that, by Prop. 1.39.(a) page 71 in [Hat], that $latex p_*\pi_1H$ is a normal subgroup of the fundamental group of . Again, by Prop. 1.39.(b), w have that the quotient is indeed .
Claim 2 Let be a connected CW complex and let be a . Then every homomorphism is induced by a map that it’s unique up to based homotopy.
Proof: This is Prop 1B.9 page 90 in [Hat]. This result can be generalised but since we need just this I will stick with this weaker version.
Now by Claim 1 we have the canonical quotient morphism which corresponds, thanks to Claim 2, to a map :
Notice that by construction
since is designed to kill . Therefore there exists a lift (Prop 1.33 page 61 in [Hat] ) which we call making the diagram commute:
Now the crucial observation is that is a -map. This is a consequence of the commutative square above and the definition (via lifting of loops) of the action on both of the total spaces.
Claim 3 We have
Proof: After observing that is an isomorphism on fibres (consequence of being a -map) one sees that fulfils the universal property of the pullback bundle, and therefore we have the claim.
Claim 4. Isomorphic principal -bundles gives rise to homotopic maps and homotopic maps gives rise to isomorphic bundles.
Proof: The second assertion is easily seen to be true, thanks to the fact that homotopic maps gives rise to isomorphic pullback bundles. For the first half, let . Being isomorphic, we have by prop 1.37 page 67 in [Hat] that , therefore the same map can be used to build $u$ which is unique up to homotopy.
This concludes the exercise, since for every (isomorphism class of) principal -bundle we construct a (homotopy class of) map classifying it.
[Hat] A. Hatcher, Algebraic Topology, Cambridge University Press.