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Let $X$ be an abstract simplicial complex. Recall that the usual simplicial chain complex for $X$ is defined as follows. Let $C_k(X)$ be the quotient of the free abelian group on formal symbols $[v_0,\ldots,v_k]$ where $\{v_0,\ldots,v_k\}$ is a $k$-simplex of $X$ which identifies $[v_0,\ldots,v_k]$ and $(-1)^{|\sigma|}[v_{\sigma(0)},\ldots,v_{\sigma(k)}]$ for all permutations $\sigma$ of $\{0,\ldots,k\}$. The differentials are defined in the usual way.

My question is what happens when you instead do the following. Let $D_k(X)$ be the free abelian group on formal symbols $[v_0,\ldots,v_k]_D$ where $\{v_0,\ldots,v_k\}$ is a simplex of $X$ of dimension at most $k$ (so possibly there are repetitions among the $v_i$). We can define a differential on $D_{\ast}(X)$ in the usual way.

To summarize : The difference between $D_k(X)$ and $C_k(X)$ are that

  1. In $D_k(X)$, the ordering matters (not just the orientation); for instance, if $v_0$ and $v_1$ are joined by an edge in $X$, then $[v_0,v_1]_D$ is unrelated to $[v_1,v_0]_D$, while in $C_1(X)$ we would have $[v_0,v_1]=-[v_1,v_0]$.

  2. In $D_k(X)$, we allow "degenerate" simplices that have repetitions in their simplices in our chain complex, while this is not allowed in $C_k(X)$.

Does $D_{\ast}(X)$ also compute the homology of $X$? I can verify that it works in degree $0$ (trivial) and $1$ (easy; but this would not work if we did not allow repetitions). It seems similar to stuff I've seen with simplicial sets, but I don't know the literature there enough to extract it from there.

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    $\begingroup$ @Albert: It seems your question is answered positively in theorem 4.3.8 of Spanier's book "Algebraic topology". It states, in your notation, that the natural map $D_\ast X\to C_\ast X$ is a quasi-isomorphism. The proof given by Spanier is a simple application of the method of acyclic models. $\endgroup$
    – Ricardo Andrade
    Commented Apr 10, 2013 at 22:57
  • $\begingroup$ By the way, let me remark that your $D_\ast(X)$ (called $\Delta(X)$ in Spanier's book) is simply the chain complex $C_\ast(S(X))$ associated with the simplicial set $S(X)$ naturally associated with the simplicial complex $X$. The $n$-simplices of $S(X)$ are your elements $[v_0,\ldots,v_n]_D$. $\endgroup$
    – Ricardo Andrade
    Commented Apr 10, 2013 at 22:58
  • $\begingroup$ You can also find this in Munkres book. $\endgroup$
    – Benjamin Steinberg
    Commented Apr 11, 2013 at 13:37
  • $\begingroup$ Indeed, the positive answer is strongly hinted at by the fact that degenerate mappings are allowed in the definition of singular homology. (Hatcher even says that the name "singular" came about because singularities are allowed in the mapping.) $\endgroup$
    – Russ Woodroofe
    Commented Oct 31, 2013 at 22:13
  • $\begingroup$ Could someone indicate how repetitions are required by Spanier? I don't see where his proof fails without them, but, as indicated, allowing repetitions is necessary for the result to be true. $\endgroup$
    – APR
    Commented Sep 5, 2019 at 21:59

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