# Simplicial set from all orderings of simplicial complex

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 $$|K|$$?

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.