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I was told there was a bijection between $[X;BG]$, the set of homotopy types of maps from a topological space $X$ to the classifying space $BG$, and the set of group homomorphisms $Hom(\pi_1 (X), G)$. But I wasn't able to find any information about this. What is the idea behind it? Could anyone give a reference?

What about, more generally, the relation between $[X,Y]$ and $Hom(\pi_1(X), \pi_1(Y))$? When is the map from left to right injective?

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    $\begingroup$ For the first question see Hatcher, Proposition 1B.9 on page 90 (it is pointed homotopy classes of maps). $\endgroup$
    – Gregory Arone
    Commented Aug 27, 2017 at 18:18
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    $\begingroup$ @MaxMustermann: Given your general question, you seem to be under the impression that $\pi_1(BG) \cong G$. This is only true if $G$ is discrete. Are you asking only about the discrete case, or in general? $\endgroup$
    – Michael Albanese
    Commented Aug 27, 2017 at 18:44
  • $\begingroup$ @MichaelAlbanese I'm interested in the discrete case first, though I would like to hear why it fails for the general case. $\endgroup$
    – Max Power
    Commented Aug 28, 2017 at 9:36
  • $\begingroup$ I am mainly using Husemoller's book, where he introduces the Milnor Construction as universal bundle in chapter 4. In Excercise 13 he asserts that the total space of a universal bundle is contractible, which implies that the basis space is Eilenberg-MacLane for discrete groups. As suggested, I would like to use e.g. Proposition 1B.9 in Hatchers book, which gives the demanded bijection only for CW complexes. But what is the cell structure of the Milnor Construction? Milnor's total space looks similar to the construction in Hatcher's book, but I don't see why the topology is the same. $\endgroup$
    – Max Power
    Commented Aug 28, 2017 at 15:56

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With regards to the second question, if $Y$ is a $K(\pi,1)$ space then the natural map from $[X,Y]$ to $Hom(\pi_{1}(X),\pi_{1}(Y))$ (divided out by conjugacy) is a bijection (see the answers here Question about maps of $S^{3}$-bundles).

But there are examples where quite the opposite occurs, take $X = Y = S^{2}$, both are simply connected but there are infinitely many non-homotopic maps.

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