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I am looking for information related to the following question:

For which $n \in\mathbb{N}$ can every group $G$ be realized as $\pi_n(X)$ for some space $X$?

I have seen in Hatcher that $n=1$ is one such case. I was wondering whether this result was true for higher $n$ as well. I tried a google search, but I didn't turn up anything, perhaps because I didn't phrase the question in an optimal manner. Is there any information out there related to this question?

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2 Answers

up vote 6 down vote accepted

For every abelian group $ G $ and $ n \geqslant 2 \; $ there is a space $ X=K(G,n)\; $ called an Eilenberg-Mac Lane space (see the index of Hatcher's book), such that $ \pi_i(X,x)=0\; $ for $i \ne n \; $ and $\pi_n(X,x) \cong G \; $.

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The short answer is: $n=1$, always; $n>1$, exactly when $G$ is abelian. This is necessary, since all higher homotopy groups are abelian, and turns out to be sufficient by the following construction. Present $G=\langle \Gamma,R\rangle$ with (possibly infinite) generating set $\Gamma$ and relations $R$. Let $X = \bigvee_\Gamma S^n$ and then for each relation $\in R$ glue in an $S^{n+1}$ according to $r$. The resulting CW-complex has $\pi_nX = G$.

You might be interested in Eilenberg-Mac Lane spaces $K(G,n)$. These are spaces which have $\pi_n(X) = G$ and all other homotopy groups zero.

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