Kerodon

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Corollary 3.2.7.7. Let $C_{\ast }$ and $D_{\ast }$ be chain complexes of abelian groups and let $f: C_{\ast } \rightarrow D_{\ast }$ be a morphism of chain complexes. The following conditions are equivalent:

$(1)$

The induced map of generalized Eilenberg-MacLane spaces $\mathrm{K}( C_{\ast } ) \rightarrow \mathrm{K}( D_{\ast } )$ is a homotopy equivalence (see Construction 2.5.6.3).

$(2)$

For every integer $n \geq 0$, the induced map of homology groups $\mathrm{H}_{n}( C ) \rightarrow \mathrm{H}_{n}(D)$ is an isomorphism.

Proof. Remark 2.5.6.4 guarantees that the simplicial sets $\mathrm{K}( C_{\ast } )$ and $\mathrm{K}(D_{\ast })$ are Kan complexes. By virtue of Theorem 3.2.7.1, $(1)$ is equivalent to the following pair of assertions:

$(1')$

The chain map $f$ induces a bijection $\pi _0( \mathrm{K}( C_{\ast } ) ) \rightarrow \pi _0( \mathrm{K}( D_{\ast } ) )$.

$(1'')$

For every vertex $x$ of $\mathrm{K}( C_{\ast } )$ having image $y \in \mathrm{K}( D_{\ast } )$ and every integer $n > 0$, the induced of homotopy groups

\[ \pi _{n}( \mathrm{K}( C_{\ast }), x) \rightarrow \pi _{n}( \mathrm{K}( D_{\ast }), y) \]

is an isomorphism.

Note that we have a commutative diagram of pointed Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ (\mathrm{K}(C_{\ast }), 0) \ar [r]^-{\mathrm{K}(f)} \ar [d]^{\sim } & ( \mathrm{K}( D_{\ast }), 0) \ar [d]^{\sim } \\ (\mathrm{K}(C_{\ast }), x) \ar [r]^-{\mathrm{K}(f)} & ( \mathrm{K}(D_{\ast }), y), } \]

where the vertical isomorphisms are given by translation by $x$ and $y$, respectively (using the group structure on the Kan complexes $\mathrm{K}( C_{\ast } )$ and $\mathrm{K}( D_{\ast } )$. Consequently, to verify $(1'')$, we may assume without loss of generality that $x = 0$. Applying Exercise 3.2.2.18, we see that $(1')$ and $(1'')$ can be reformulated as follows:

$(2')$

The chain map $f$ induces an isomorphism $\mathrm{H}_{0}( C ) \rightarrow \mathrm{H}_0(D)$.

$(2'')$

For every integer $n > 0$, the chain map $f$ induces an isomorphism $\mathrm{H}_{n}(C) \rightarrow \mathrm{H}_ n(D)$.

$\square$