# Homology of simplicial set vs homology of its geometric realization?

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?

• Spanier has all the details on the funtorial isomorphism between simplicial homology and singular homology of the geometric realization.
– Pedro
Sep 29, 2016 at 9:21
• A fun way of seeing the isomorphism if you are familiar with spectral sequences is to filter the geometric realization by skeleta and look and the induced spectral sequence of a filtered space. It converges against the singular homology of the geometric realization and the E_1-term is concentrated in the 0th row, where it is given by the simplicial chain complex. Oct 1, 2016 at 13:09
• @archipelago I just found your comment, can you explain how you see that the $0$th row is the simplicial chain complex ? The spectral sequence I know has $E^1_{p,0} = H_p(|X|^{(p)}, |X|^{(p-1)})$ which I can see is the cellular chain complex, but I have trouble relating it to the simplicial one (if I instead filter the simplicial complex, I get nothing interesting) Mar 28, 2019 at 16:36

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.