Main question is this: Suppos $M$ is a manifold and $G$ is a finite group. If there is a group homomorphism $\phi:\pi_1 M\to G$, is there a continuous map $f:M\to BG$, where $BG$ is a classifying space?
One may generalize this question to ask if the functor $\pi_1$ is full, but I think this is too good to be true.
This question arises from the following context. When I was reading some article, I read that there is a bijection between the isomorphism class of principal $G$-bundles over $M$ and the homotopy class of maps from $M$ to $BG$. I wanted to verify this, so I found out that (1) if two maps $f,g:M\to BG$ are in the same homotopy class, $f_*, g_*:\pi_1 M\to G$ are conjugate, (2) a map $M\to BG$ induces a principal $G$-bundle, which is a pull back of $EG\to BG$, (3) if $f_*, g_*$ are in the same conjugacy class, the principal $G$-bundles induced by each of them are isomorphic to each other, where the isomorphism is $p\mapsto hph^{-1}$ where $hf_*h^{-1}=g$, and (4) a principal $G$-bundle $N\to M$ induces a group homomorphism $\pi_1(M)\to G$, since every loops on $M$ induces a $G$-action on $M$. I can deduce that the homotopy class of maps from $M$ to $BG$ gives an isomorphism class of principal $G$-bundles over $M$, but I am stuck on the converse statement. I think it is enough to prove that $\pi_1(M)\to M$ gives a continuous map $M\to BG$, and I presume that there is something with classifying space.
Thanks in advance.
Best Answer
Yes, such a map $f$ does exist. In fact, it exists even when $M$ is a CW complex (and every smooth manifold has a CW complex structure) and when $G$ is an arbitrary group (not just a finite groups).
The proof is a straightforward obstruction theory argument. First pick a maximal tree in the 1-skeleton of $M$ and map that to the base point of $BG$. Next, each remanining 1-cell in $M$ represents an element of $\pi_1(M)$, map that edge to a loop in $BG$ representing the $\phi$ image of that element. Now extend over the 2-skeleton: the boundary of each 2-cell represents the trivial element of $\pi_1(M)$, so the image loop represents the trivial element of $\pi_1(BG)$, so you can extend the map continuously over that 2-cell. Now induct up the dimensions: assuming the map is defined on the $n$-skeleton for $n \ge 2$, for each $n+1$ cell the restriction of the map to the boundary of that cell is homotopically trivial, because $\pi_n(BG)$ is trivial, so the map can be extended continuously over the whole $n+1$ cell.