Identifying the differentials of the cellular chain complex of a (fat) geometric realization

Question

Let $X_\bullet$ be a simplicial set, $\|X_\bullet\|$ its fat geometric realization and $|X_\bullet|$ its geometric realization, see here for definitions.

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The fat geometric realization $\|X_\bullet\|$ has a natural CW complex structure with $n$ cells in bijection with $X_n$ and attaching maps done in accordance with face maps. One thus expects:

Plausible fact 1. The Moore complex $CX_\bullet$ is naturally isomorphic with the cellular homology complex of this CW complex structure on $\|X_\bullet\|$.

And indeed one can find this fact stated in Dupont's Curvature and Characteristic Classes at the beginning of the proof of Proposition 5.15:

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The geometric realization $|X_\bullet|$ also has a CW complex structure with $n$ cells in bijection with the nondegenerate simplices of $X_n$ (Proposition 2.3 in Goerss-Jardine) and with attaching maps done in accordance with face maps. One similarly expects that the quotient complex $CX_\bullet/DX_\bullet$, where $DX_\bullet$ is the subcomplex generated by all degenerate simplices, is the cellular homology complex of the natural CW complex structure on $|X_\bullet|$. Also, since the normalized chain complex $NX_\bullet$ is naturally isomorphic to $CX_\bullet/DX_\bullet$ one expects

Plausible fact 2. The normalized Moore complex $NX_\bullet$ is naturally isomorphic to the cellular homology complex of this CW complex structure on $|X|$.

And indeed one finds this statement as Proposition 13.8 in P. May's Classifying Spaces and Fibrations

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Both points above are clearly true of the underlying graded abelian groups. The sources I cite further claim that this is true of the differentials involved. This seems utterly believable since in both cases the attaching maps of are "done in accordance with face maps" and it seems attaching is done "without multiplicities" in the sense that all attaching maps either "collapse something" and should have degree $0$ or are affine homeomorphisms onto their image and should have degree $\pm1$. Neither fact is given a full proof, however.

Question. Is there a nice write-up somewhere of these facts? I.e. of the computations identifying the differentials of the cellular homology complexes of $\|X_\bullet\|$ and $|X_\bullet|$ with those of the Moore complex and the normalized Moore complex respectively? Alternatively if it's not too much work could someone spell it out?

Answer 1

The semi-simplicial case. This is a partial answer that should contain most of what is needed for a clean proof. The proof for the normalized case should be very similar, although slightly complicated by the fact that some simplices will be "modded out". I'd still like a clean reference where the details are worked out! A good reference to a textbook/paper that treats either or both of the differentials will get the bounty.

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A fact that ought to be true

Let $X_\bullet$ be a semi-simplicial set, i.e. a contravariant functor $\Delta_\text{inj}\to\mathrm{Set}$. Let $\Delta^\bullet$ be the usual co-semi-simplicial topological space with $\Delta^\bullet[n]=\Delta^n$ the standard $n$-simplex and the $d^i:[p]\to[q]$ realized by the usual affine maps. I'll take for granted that

Fact. $$ \|X_\bullet\| = \int^n\Delta^n\times X_n = \mathrm{coeq}\bigg( \bigsqcup_{\varphi:[p]\hookrightarrow[q]} \Delta^p\times X_q\rightrightarrows\bigsqcup_{[r]} \Delta^r\times X_r \bigg) $$ is a CW complex with the filtration given by the $$ \|X_\bullet\|^{(n)} = \mathrm{coeq}\bigg( \bigsqcup_{\substack{\varphi:[p]\hookrightarrow[q]\\q\leq n}} \Delta^p\times X_q\rightrightarrows\bigsqcup_{r\leq n} \Delta^r\times X_r \bigg) $$ and that the topology works out perfectly. In particular that there are pushout diagrams $$ \begin{array}{ccc} \displaystyle\bigsqcup_{X_n} \partial\Delta^n & \hookrightarrow & \displaystyle\bigsqcup_{X_n} \Delta^n \\ \downarrow & &\downarrow\\ \|X_\bullet\|^{(n-1)} & \hookrightarrow & \|X_\bullet\|^{(n)} \end{array} $$ for the canonically defined maps $$ \bigsqcup_{\alpha\in X_n} \phi_{\alpha}:\bigsqcup_{X_n} \Delta^n\longrightarrow\|X_\bullet\|^{(n)} $$ That arise in the coequalizers above, i.e.: $$ \bigsqcup_{\alpha\in X_n} \phi_{\alpha}:\bigsqcup_{X_n} \Delta^n \hookrightarrow \bigsqcup_{r\leq n} \Delta^r\times X_r \xrightarrow{~\mathrm{coeq}~} \|X_\bullet\|^{(n)} $$ Also $\|X_\bullet\|$ has the weak topology relative to this filtration.

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Setup

I guess to make the claim about the differentials work it is enough to slightly modify the cellular chain complex of $\|X_\bullet\|$. I.e. rather than use maps $D^{n}\to \|X_\bullet\|^{(n)}$ which restrict to maps $\partial D^{n}\to \|X_\bullet\|^{(n-1)}$, where $D^n$ is the euclidean radius 1 ball in $\Bbb{R}^n$, we use the maps $\Delta^{n}\to\|X_\bullet\|^{(n)}$ implicit in the coequalizer. In particular this gives us the following picture

The cyan $\color{cyan}{\sim}$ are isomorphisms in homology by general properties of homology. We endow every relative homology group $H_p(\Delta^p,\partial \Delta^p)$ with the basis element $[\mathrm{id}_{\Delta^p}]$. This provides us with an isomorphism of abelian groups $$ \begin{array}{ccccc} \displaystyle \big(CX_\bullet\big)_n & \simeq & \displaystyle \bigoplus_{X_{n}}H_n(\Delta^n,\partial\Delta_n) & \displaystyle\overset{H(\bigsqcup_{\alpha\in X_n}\phi_\alpha)}\simeq & \displaystyle H_n(\|X_\bullet\|^{(n)}, \|X_\bullet\|^{(n-1)}) \\ \end{array} $$ Under the connecting homomorphism this basis vector becomes the basis vector $\partial[\mathrm{id}_{\Delta^p}]=\sum_{i=0}^p(-1)^i[\mathrm{id}_{\Delta^p}\circ d^i]=\sum_{i=0}^p(-1)^i[d^i]$ using the structure maps of the co-semi-simplicial set $\Delta^\bullet$.

Now to conclude it will be enough to note that for every $(n+1)$-simplex $\alpha\in X_{n+1}$ and its associated attaching map $\phi_\alpha:\Delta^{n+1}\to\|X_\bullet\|^{(n+1)}$ the two paths below are the same:

Using the commutativity of the first diagram one can alternatively show that the orange and purple paths below define the same map:

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Computation

Let $\alpha_0\in X_{n+1}$ be an $n+1$-simplex. The orange path does the following:

1. Compute the Moore differential of the basis element $\alpha_0\in (CX_\bullet)_{n+1}$, i.e. $$\alpha_0\mapsto\sum_{i=0}^{n+1}(-1)^id_i(\alpha_0).$$
2. Send it to the following sum $$\sum_{i=0}^{n+1}(-1)^id_i(\alpha_0)\mapsto\sum_{i=0}^{n+1}(-1)^i [id_{\Delta^n}]_{d_i(\alpha_0)}$$ where $[id_{\Delta^n}]_{d_i(\alpha_0)}$ designates the previously highlighted basis vector of the $d_i(\alpha_0)$-th copy of $H_n(\Delta^n,\partial\Delta_n)$ in $\bigoplus_{X_n}H_n(\Delta^n,\partial\Delta^n)$.
3. Compute its image under the map induced in homology $\displaystyle \Big(\bigsqcup_{\beta\in X_n}\phi_\beta\Big)_*$ of the attaching maps in dimension $n$: $$ \begin{array}{rcl} \displaystyle \sum_{i=0}^{n+1}(-1)^i [id_{\Delta^n}]_{d_i(\alpha_0)} & \mapsto & \displaystyle \sum_{i=0}^{n+1}(-1)^i \Big(\bigsqcup_{\beta\in X_n}\phi_\beta\Big)_*[id_{\Delta^n}]_{d_i(\alpha_0)} \\ & & \displaystyle \quad = \sum_{i=0}^n(-1)^i \Big[\phi_{d_i\alpha_0}:\Delta^n \to X_n \Big] \\ & & \displaystyle \quad = \sum_{i=0}^n(-1)^i \Big[\phi_{d_i\alpha_0} \Big] \end{array} $$

On the other hand, the purple path does the following:

1. send $\alpha_0\in X_{n+1}$ to the element $[\mathrm{id}_{\Delta^{n+1}}]\in H_{n+1}(\Delta^{n+1},\partial\Delta^{n+1})$ embedded at the $\alpha_0$-th summand in $\bigoplus_{X_{n+1}}H_{n+1}(\Delta^{n+1},\partial\Delta^{n+1})$: $$ \alpha_0\mapsto [\mathrm{id}_{\Delta^{n+1}}]\in H_{n+1}(\Delta^{n+1},\partial\Delta^{n+1}) \overset{\iota_{\alpha_0}}\hookrightarrow \bigoplus_{X_{n+1}}H_{n+1}(\Delta^{n+1},\partial\Delta^{n+1}) $$ let's call it $[\mathrm{id}_{\Delta^{n+1}}]_{\alpha_0}$
2. Apply the connecting homomorphism to $[\mathrm{id}_{\Delta^{n+1}}]_{\alpha_0}$, i.e. $$ \begin{array}{rcl} [\mathrm{id}_{\Delta^{n+1}}]_{\alpha_0} & \mapsto & \sum_{i=0}^n(-1)^i [d^i:\Delta^{n}\hookrightarrow \Delta^{n+1}]_{\alpha_0} \\ & &\quad = \sum_{i=0}^n(-1)^i [d^i]_{\alpha_0} \end{array} $$
3. Apply the map induced in homology $\Big(\bigsqcup_{\alpha\in X_{n+1}}\phi_\alpha\Big)_*$: $$ \begin{array}{rcl} \displaystyle\sum_{i=0}^n(-1)^i [d^i]_{\alpha_0} & \mapsto & \displaystyle \sum_{i=0}^n(-1)^i \Big(\bigsqcup_{\alpha\in X_{n+1}}\phi_\alpha\Big)_* [d^i]_{\alpha_0} \\ & & \displaystyle \quad = \sum_{i=0}^n(-1)^i \Big[\Delta^n \xrightarrow{d^i} \Delta^{n+1} \xrightarrow{\phi_{\alpha_0}} X_{n+1} \Big] \\ & & \displaystyle \quad = \sum_{i=0}^n(-1)^i \Big[\phi_{\alpha_0}\circ d^i \Big] \end{array} $$

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Conclusion

Thus, it would be sufficient if all the the attaching maps $\phi_\alpha$, $\alpha\in X_{n+1}$ satisfy, for all $\alpha\in X_{n+1}$ and all $i\in[n+1]$, $$\phi_\alpha\circ d^i=\phi_{d_i\alpha}$$ Which I believe will be the case by virtue of the Fact above and the way the maps $\phi_\alpha$, $\alpha\in X_n$ are defined.