Homology of simplicial set vs homology of its geometric realization?

Question

Problem. Given a simplicial set $X$, we can associate to it a chain complex defined by $$C_n=\mathbb{Z}[X_n]$$(the free abelian group on the set $X_n$) with differentials $$d=\sum_i(-1)^id_i:C_n\to C_{n-1},$$ the alternating sum of the face maps. Is the homology of this chain complex the same as the singular homology of the geometric realization, i.e. do we have $$H(C_*)\cong H\left(|X|;\mathbb{Z}\right)\quad ?$$

Attempt. There is a natural map on the level of chain complexes given by

$$\begin{align}C_n \longrightarrow C_n^{sing}\left(|X|\right)\\ x_n\mapsto\left(t\mapsto[t,x_n]\right)\end{align}$$

and I wonder if this induces an isomorphism on homology.

Alternatively, $|X|$ is a CW complex with one $n$-cell $e_n^i$ for each nondegenerate $n$-simplex $x_n^i$, so we can look at the chain map

$$\begin{align} C_n^{cell}\left( |X| \right) \longrightarrow C_n\\ e_n^i\mapsto x_n^i \end{align}$$

Is this a homology isomorphism?

Answer 1

The answer is yes. Let me first establish some notation. Let $X$ be a simplicial set. If $Y$ is a space, let $S_\bullet(Y)$ be the singular simplicial set. Given a simplicial abelian group $M$, write $C_\bullet(M)$ to be your alternating chain map chain complex.

The natural map on the level of chain complexes you've written comes from the simplicial map: $$ X \hookrightarrow S_\bullet( \vert X \vert). $$ First apply $\mathbb Z[-]$, which gives you a simplicial abelian group morphism, then take the induced map on the alternating face map chain complexes: $$ C_\bullet \mathbb Z[X] \to C_\bullet \mathbb Z[S_\bullet( \vert X \vert)]. $$ Taking homology of this gives $$ H_\bullet(C_\bullet(\mathbb Z[X])) \to H_\bullet^{\text{sing}}(\vert X \vert). $$

One way to see this, due to Milnor ("Geometric Realization of Semi-Simplicial Complexes," 1956), is as follows.

Note that $\vert X \vert$ is a CW-complex and there is a correspondence between the CW-filtration and the skeletal filtration of $X$, that is, $\vert \text{sk}_n(X) \vert \cong \vert X \vert_n$.

The CW-filtration,

$$\emptyset \hookrightarrow \vert X \vert_0 \hookrightarrow \vert X \vert_1 \hookrightarrow \vert X \vert_2 \hookrightarrow \cdots \hookrightarrow \vert X \vert$$

is a tower of cofibrations, which gives rise to a spectral sequence with

$$E_{p,q}^1 = H_{p+q}^{\text{sing}}(\vert X \vert_p ,\vert X \vert_{p - 1}) \Longrightarrow H_{p+q}^{\text{sing}}(\vert X \vert).$$

On the other hand, the chain complex $C_\bullet \mathbb Z[X]$ has a natural filtration of chain complexes:

$$C_\bullet \mathbb Z [\text{sk}_0(X)] \hookrightarrow C_\bullet \mathbb Z [\text{sk}_1(X)] \hookrightarrow C_\bullet \mathbb Z [\text{sk}_2(X)] \hookrightarrow \cdots \hookrightarrow C_\bullet \mathbb Z [X]$$

which gives rise to a spectral sequence with

$$\overline E_{p,q}^1 = H_{p+q}(\text{nd}_p(X)) \Longrightarrow H_{p+q}(C_\bullet \mathbb Z [X]),$$

where $\text{nd}_p(X)$ are the non-degenerate $p$-simplices of $X$.

The key observation now is that the simplicial map $X \hookrightarrow S_\bullet(\vert X \vert)$ induces an isomorphism $\overline E_{p, q}^1 \to E_{p, q}^1$ with naturality guaranteeing that this yields an isomorphism of spectral sequences, hence they must converge to the same things.