Understanding a step in example 4.34 in Hatcher.

Question

In example 4.34, Hatcher shows that for $n > 1$, a CW Moore space $M(G,n)$ has a unique homotopy type for given abelian $G$ and $n$. He uses a given $M(G,n)$, $X$, constructed by attaching $(n+1)$ cells to a wedge sum of $n$-spheres. He then lets $Y$ be any other $M(G,n)$ CW complex.

Then he applies lemma 4.31 to obtain a map $f: X \to Y$ inducing an isomorphism on $\pi_n$. This lemma states that for $X$ constructed by attaching $(n+1)$ cells to a wedge sum of $n$ spheres, then for any homomorphism $\pi_n(X) \to \pi_n(Y)$, there exists a map $f: X \to Y$ inducing said homomorphism on $\pi_n$.

It appears that Hatcher is using the fact that $\pi_n(X)$ and $\pi_n(Y)$ are isomorphic. I fail to see why this is the case. I believe it's because we can apply the Hurwicz Theorem to $Y$, which would tell us that $\pi_n(Y) \cong H_n(Y) = G$, but a hypothesis is that $Y$ be $(n-1)$ connected, which I don't believe we can assume. Am I missing something?

Answer 1

First note that Hatcher takes Moore spaces $M(G, n)$ to be simply connected if $n > 1$. So suppose $Y$ is simply connected and $\widetilde{H}_i(Y) = 0$ if $i \neq n$ and $H_i(Y) = G$.

Let $k$ be the first positive integer such that $\pi_k(Y) \neq 0$. As $Y$ is simply connected $k > 1$. By the Hurewicz theorem, $H_k(Y) = \pi_k(Y) \neq 0$ so we must have $k = n$ and hence $Y$ is $(n-1)$-connected.