Let $G$ be a Lie group and $P \to M$ a principal $G$-bundle over a closed Riemann surface. The gauge group $\mathcal{G}$ is defined by

$$\mathcal{G}=\lbrace f : P \to G \mid f(p \cdot g) = g^{-1}f(p)g\ (\forall g \in G, p \in P) \rbrace.$$

I want to understand the following statement which I found in Atiyah-Bott (Roy. Soc. London Ser. A 308).

PROPOSITION 2.4. Let $BG$ be the classifying space for $G$. Then in homotopy theory $$B\mathcal{G} = \mathrm{Map}_P(M,BG).$$ Here the subscript $P$ denotes the component of a map of $M$ into $BG$ which induces $P$.

This is proved by showing that $\pi:\mathrm{Map}_G(P,EG) \to \mathrm{Map}_P(M,BG)$ is a universal $\mathcal{G}$-bundle.

My questions:

- Why is $\mathrm{Map}_G(P,EG)$ contractible?
- How do we see that $\pi:\mathrm{Map}_G(P,EG) \to \mathrm{Map}_P(M,BG)$ is locally trivial? Here for a $G$-equivariant map $f:P \to EG$, $\pi(f)$ is the map $P/G \to EG/G$ which is canonically induced by $f$.

Notes:

It is easy to see that $\mathrm{Map}(P,EG)$ is contractible, because $EG$ is contractible. But, because of the $G$-actions, it does not seem that we can apply the same discussion to $\mathrm{Map}_G(P,EG)$.

For the second question, Atiyah and Bott say the following. But I can not understand it and I want to understand the details.

If $BG$ is paracompact and locally contractible, which is easily arranged, $\pi$ will be a locally trivial principal fibring, as follows easily from the homotopy properties of fibrings.

If there is a good exposition of this problem, please let me know.