Proposition 3.5.3.9. Let $(A_{\ast }, \partial )$ be a chain complex of abelian groups, let $X = \mathrm{K}( A_{\ast } )$ be the associated Eilenberg-MacLane space, and let $n$ be a nonnegative integer. Then $X$ is $n$-coskeletal if and only if it satisfies the following conditions:
- $(a)$
The abelian groups $A_{m}$ vanish for $m \geq n+2$.
- $(b)$
The boundary map $\partial : A_{n+1} \rightarrow A_{n}$ is a monomorphism, whose image is the group of $n$-cycles $Z_{n} = \{ y \in A_{n}: \partial y = 0 \} $.
Proof.
Fix an integer $m \geq 0$, and let $\sigma : \Delta ^ m \rightarrow \Delta ^ m$ be the identity map, which we identify with its image in the normalized chain complex $\mathrm{N}_{\ast }( \Delta ^ m; \operatorname{\mathbf{Z}})$. Then $\partial (\sigma ) = \sum _{i=0}^{m} (-1)^{i} d^{m}_{i}(\sigma )$ is a cycle in the subcomplex $\mathrm{N}_{\ast }( \operatorname{\partial \Delta }^{m}; \operatorname{\mathbf{Z}})$. Suppose we are given a morphism of simplicial sets $\tau _0: \operatorname{\partial \Delta }^{m} \rightarrow X$, which we identify with a chain map $f_0: \mathrm{N}_{\ast }( \operatorname{\partial \Delta }^{m}; \operatorname{\mathbf{Z}}) \rightarrow A_{\ast }$. Then $y = f_0( \partial \sigma )$ is an $(m-1)$-cycle of $A_{\ast }$. Note that, if $m > 0$, then every $(m-1)$-cycle $y$ of $A_{\ast }$ can be obtained in this way (for example, we can take $f_0: \mathrm{N}_{\ast }( \operatorname{\partial \Delta }^{m}; \operatorname{\mathbf{Z}}) \rightarrow A_{\ast }$ to be the map of chain complexes which carries $d^{m}_{0}( \sigma )$ to $y$ and every other nondegenerate simplex to zero).
If $\tau : \Delta ^{m} \rightarrow X$ is an extension of $\tau _0$ corresponding to a map of chain complexes $f: \mathrm{N}_{\ast }( \Delta ^{m}; \operatorname{\mathbf{Z}}) \rightarrow A_{\ast }$, then $x = f( \sigma )$ is an $m$-chain of $A_{\ast }$ satisfying $\partial (x) = y$. This construction induces a bijection
\[ \{ \textnormal{$m$-simplices $\tau $ with $\tau |_{ \operatorname{\partial \Delta }^{m} } = \tau _0$} \} \xrightarrow {\sim } \{ \textnormal{$x \in A_{m}$ with $\partial (x) = y$} \} . \]
It follows that the simplicial set $X$ is $n$-coskeletal if and only if it satisfies the following condition for each $m > n$:
- $(b_ m)$
The boundary map $\partial : A_{m} \rightarrow A_{m-1}$ is a monomorphism whose image is the group of $(m-1)$-cycles $Z_{m-1} = \{ y \in A_{m-1}: \partial y = 0 \} $.
Note that $(b_{n+1})$ is a restatement of $(b)$. Moreover, if condition $(b_ m)$ is satisfied for some integer $m$, then condition $(b_{m+1} )$ is equivalent to the requirement that the abelian group $A_{m+1}$ is trivial. In particular, $(b_ m)$ is satisfied for all $m > n$ if and only if $A_{\ast }$ satisfies conditions $(a)$ and $(b)$.
$\square$