Exercise 3.2.2.22 (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 3.2.2.21).
- $(2)$
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