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 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.
Claim 3 The map is a -map.
We have to prove the following:
for every and every . Since is a -bundle, hence (with our assumptions) a normal cover, it’s easy to observe that we can express the action of via lifts of appropriate loops on . Therefore let be a loop representing . Let be the lift of based at . Similarly let be a loop representing , and let be the (unique) lift of it starting at . With this notation what we need to prove is the following
Notice that is the map inducing the isomorphism
hence . Notice now that since is a lift of and is a lift of at the same basepoint (in fact by construction ). Hence after lifting the homotopy on we notice that the two lifts must be homotopy equivalent rel. end points, in particular their value at must coincide, which is precisely what we need here.
Claim 4 The pullback of along the map is isomorphic to the bundle . In symbols:
Proof: Consider the pullback of along . It’s easy to observe that we have the following commutative diagram using the universal property of the pullback (in the category of -bundles.
Notice that is a lift of , and for the same reason given for , it’s a -map. By the classification of coverings of ([Hat] Thm. 1.38 page 67) is an isomorphism of bundle and a -map, hence an isomorphism of principal -bundles. Hence the claim
Claim 5. 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.