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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 $X$

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2 Answers 2

up vote 7 down vote accepted

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 $\pi_1Map(X,K(A,n))=[\Sigma X,K(A,n)]=H^n(\Sigma X,A)\cong H^{n-1}(X,A)$.

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Perfect! so also $\pi_jMap(X,K(A,n))\cong H^{n−j}(X,A)$ for $j\leq n$, which is precisely what I needed. thanks a lot! –  domenico fiorenza Jul 5 '11 at 21:36

One can sharpen Fernandos answer: there exist models for Eilenberg Mac Lane spaces that are abelian topological groups. Hence the mapping space $Map(X;K(A,n))$ (based or unbased) is a topological abelian group. Topological abelian groups are products of Eilenberg-Mac Lane spaces, and so your mapping space is a product of Eilenberg-Mac Lane spaces, too. Which means that you know the Postnikov tower as well.

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