Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 3.1.5.17. Let $f: X \rightarrow Y$ be a weak homotopy equivalence of simplicial sets. Then the induced map of normalized chain complexes $\mathrm{N}_{\ast }(X; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{N}_{\ast }(Y; \operatorname{\mathbf{Z}})$ is a chain homotopy equivalence. In particular, $f$ induces an isomorphism of homology groups $\mathrm{H}_{\ast }(X;\operatorname{\mathbf{Z}}) \rightarrow \mathrm{H}_{\ast }(Y; \operatorname{\mathbf{Z}})$.

Proof. Let $M_{\ast }$ be a chain complex of abelian groups. We wish to show that precomposition with $\mathrm{N}_{\ast }(f; \operatorname{\mathbf{Z}})$ induces a bijection

\[ \xymatrix { \{ \text{Chain homotopy classes of maps $\mathrm{N}_{\ast }(Y; \operatorname{\mathbf{Z}}) \rightarrow M_{\ast }$} \} \ar [d]^{\theta } \\ \{ \text{Chain homotopy classes of maps $\mathrm{N}_{\ast }(X; \operatorname{\mathbf{Z}}) \rightarrow M_{\ast }$} \} . } \]

Let $\mathrm{K}(M_{\ast })$ denote the Eilenberg-MacLane space associated to $M_{\ast }$ (Construction 2.5.6.3). Using Example 3.1.4.8, we can identify $\theta $ with the map

\[ \pi _{0}(\operatorname{Fun}(Y, \mathrm{K}(M_{\ast } ) )) \rightarrow \pi _0( \operatorname{Fun}(X, \mathrm{K}(M_{\ast } ) ) ) \]

given by precomposition with $f$. This map is bijective because $f$ is a weak homotopy equivalence (by assumption) and $\mathrm{K}(M_{\ast })$ is a Kan complex (Remark 2.5.6.4). $\square$