Let $X$ be a finite CW-complex and $A$ an abelian group, and consider the space $Maps(X,K(A,n))$ of continuous maps from $X$ to $K(A,n)$ endowed with the compact-open topology, so that it represents the functor $Y\mapsto C(X\times Y,K(A,n))$. Let $Maps\_0(X,K(A,n))$ be the path-connected component of $Maps(X,K(A,n))$ corresponding to the zero in $H^n(X,A)$, and let $\*$ any point in $Maps\_0(X,K(A,n))$. I'm interested in the fundamental group $\pi\_1( Maps\_0(X,K(A,n)),\*)$.


i) is it abelian?


ii) does it act trivially on $\pi\_i( Maps\_0(X,K(A,n)),\*)$ for $i\geq 1$?


If the answer to these two were "yes", then one would have that for $X$ a closed compact connected smooth oriented manifold of dimension $k\lt n$, the Postnikov tower of $Maps(X,K(\mathbb{Z},n))$ begins with $K(\mathbb{Z},n-k)$, so that there would be a canonical $(n-k-2)$-gerbe on the (moduli) space of $(n-2)$-gerbes on $X$



Fundamental groups of spaces of maps to Eilenberg-MacLane spaces
In general, the homotopy groups (based at the trivial map) of a **based** mapping space $Map(X,Y)$ are $\pi\_nMap(X,Y)=[\Sigma^n,Y]$, $n\geq 0$, where the brackets denote sets of homotopy classes of maps. If $Y=K(A,n)$ then you get $\pi\_1Map(X,K(A,n))=[\Sigma X,K(A,n)]=H^n(\Sigma X,A)\cong H^{n-1}(X,A)$.


