Given an abstract simplicial complex $K$ on a set of vertices $V$, we can form a semi-simplicial set by $F(K)$ sending $F(K)_n$ to be the set of ordered $(n+1)$-tuples of vertices in $V$ forming an $n$-simplex in $K$; face maps are defined in the obvious way. One can add in degeneracies by hand to make this a simplicial set.

In general, $|F(K)|$ is not homotopy equivalent to $K$; for example, $|F(\Delta^n)|$ is a wedge of $n$-spheres. (By contrast, the "singular simplicial set" where we don't require the $(n+1)$-tuple to form a $n$-simplex but merely any simplex, is homotopy equivalent and even homeomorphic in the geometric realization to $|K|$: see this question) This I learned from the discussion here. However, I wanted to know what information in the literature exists about the relationship between $|K|$ and $|F(K)|$ in general: for example, are there some restrictions on the homology or homotopy of $|F(K)|$ based on the homology or homotopy of $|K|$?


1 Answer 1


If I understand this correctly, this construction shows up in various places in the study of homological stability. The one main result I know connecting the topology of $F(K)$ and $K$ is as follows. Say that $K$ is weakly Cohen-Macaulay of dimension $n$ if $K$ is $(n-1)$-connected and for all simplices $\sigma$ of $K$, the link of $\sigma$ is $(n-2-dim(\sigma))$-connected. For instance, it is easy to see that this holds if $K$ is a combinatorial triangulation of an $(n-1)$-connected manifold of dimension at least $n$ (the "dimension at least $n$" is here because I'm silently using the convention that only nonempty spaces can be $p$-connected for $p \geq -1$). We then have the following theorem:

Theorem: Let $K$ be a simplicial complex that is weakly Cohen-Macaulay of dimension $n$. Then $F(K)$ is $(n-1)$-connected.

I think this should be attributed to Randal-Williams--Wahl; see Theorem 2.14 of their paper here. An alternate proof by Hatcher--Vogtmann can be found as Proposition 2.10 in their paper here.

  • $\begingroup$ this is a great result, thank you so much! $\endgroup$
    – xir
    Mar 14 at 1:09

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