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

for

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).

**Claim 1**

**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.

**Reference:**

**[Hat]** A. Hatcher, *Algebraic Topology*, Cambridge University Press.