Fundamental group of the space of maps into a classifying space Let $P : E \to X$ be a principal $G$-bundle, where $G$ is a connected topological group. $P$ is classified by a map $f: X \to BG$. The group of gauge transformations $\mathcal{G}$ of $P$ is defined to be the group of $G$-equivariant homeomorphisms $f : E \to E$ over $X$. With the usual notations, in homotopy we have:
$B\mathcal{G} \simeq Map_{f}(X,BG)$
I am wondering, what is known about the homotopy groups $\pi_{k}(B\mathcal{G})$ when $X$ has dimension bigger than four and $G=SU(n)$? In particular, it is known if $B\mathcal{G}$ is simply connected?
Thanks.
 A: Very little is know about these spaces. Most of the literature focuses on studying the homotopy of the gauge groups of principal $G$-bundles with $G$ a simply connected, compact Lie group and $X$ a $4$-sphere or other simply connected $4$-manifold. And these alone are very delicate problems. Very little is known about these gauge groups when $X$ is of larger dimension.
For an idea of the complexity of the problem see "Homotopy type of gauge groups of quaternionic line bundles over spheres" by Claudio and Spreafico. They look at $SU(2)$-gauge groups over spheres of various dimensions - and even here it is a case-by-case problem!
You would do well to keep track of the work of Stephen Theriault and his students, whom I know are interested in working with more intricate base spaces $X$. I know there is work being done with $X$ a 2-connected $7$-manifold.
There is one solution, whose application and finer details I will leave you to decide. For the classical groups $G=SO(n),SU(n),Sp(n),Spin(n)$ and so on you have the infinite dimensional groups $SO(\infty),SU(\infty)$, and so on. Let us assume for simplicity that $G=SU=SU(\infty)$. The same reasoning applies to all such groups. Then the gauge groups $\mathcal{G}(X,SU)$ of these infinite dimensional groups are all trivial. This follows since one may use the $H$-structure of $BSU$ to constuct a section to the evaluation fibration $ev:B\mathcal{G}\simeq Map_f(X,BSU)\rightarrow BSU$. Similarly we may use the induced $H$-structure to show that all the components of $Map(X,BSU)$ have the same homotopy type. Thus $B\mathcal{G}\simeq BSU\times Map_*^0(X,BSU)$, where $Map_*^0(X,BSU)$ is the component of the constant map in the space of pointed maps $X\rightarrow BSU$. Thus $\pi_kB\mathcal{G}=\pi_kBSU\oplus K^k(X)$.
Now for each $n$ the map classifying the inclusion $Bj_n:BSU(n)\rightarrow$ is $(2n+1)$-connected. One sees that $\pi_kB\mathcal{G}(X,SU(n))$ stabilizes as $n$ grows (analogous to Bott periodicity). Using the above construction then, for a given $n$ one readily obtains information about $\pi_kB\mathcal{G}(X,SU(n))$ in a range $0<k<(2n+1-\text{dim}(X))$.
For general results this is about the best that one can do. With the current technology it is most a case-by-case study of these objects with respect to the differing complexity of $X$ or $G$.
