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Exercise 3.2.2.18 (Homotopy of Eilenberg-MacLane Spaces). Let $M_{\ast }$ be a chain complex of abelian groups and let $X = \mathrm{K}(M_{\ast })$ be the associated Eilenberg-MacLane space (Construction 2.5.6.3). Let $x \in X$ be the vertex corresponding to the zero element, and let $n$ be a positive integer. Note that a pointed map from $\Delta ^ n / \operatorname{\partial \Delta }^ n$ to $X$ can be identified with a map of chain complexes $\mathrm{N}_{\ast }( \Delta ^ n, \operatorname{\partial \Delta }^ n; \operatorname{\mathbf{Z}}) \simeq \operatorname{\mathbf{Z}}[n] \rightarrow M_{\ast }$: in other words, it can be identified with an $n$-cycle of the chain complex $M_{\ast }$, which we will denote by $\overline{\sigma }$.

$(1)$

Let $\sigma , \sigma ': \Delta ^ n \rightarrow X$ be $n$-simplices whose restriction to $\operatorname{\partial \Delta }^ n$ is equal to the constant map $\operatorname{\partial \Delta }^ n \rightarrow \{ x\} \hookrightarrow X$. Show that $[\sigma ] = [\sigma ']$ in $\pi _{n}(X,x)$ if and only if $\overline{\sigma }$ and $\overline{\sigma }'$ are homologous as $n$-cycles of $M_{\ast }$ (use Remark 3.2.2.17).

$(2)$

Show that the $[\sigma ] \mapsto [\overline{\sigma }]$ induces an isomorphism from $\pi _{n}(X,x)$ to the homology group $\mathrm{H}_{n}(M)$.

In particular, if $A$ is an abelian group and $m \geq 0$ is an integer, then the homotopy groups of the Eilenberg-MacLane space $X = \mathrm{K}(A,m)$ are given by

\[ \pi _{n}(X,x) = \begin{cases} A & \text{ if } n=m \\ 0 & \text{ otherwise. } \end{cases} \]