A class of simplicial complexes defined by arithmetic properties

Question

The purpose of my question is to ask about properties in a certain class of 3-dimensional (and other odd dimensional) simplicial complexes. I will first describe the construction in 3 dimension and then in general odd dimensions.

Let $n,g$ and $h$ be integers. Consider the pure simplicial complex $K(n;g,h)$ on the vertex set $[n]=\{1,2,\dots,n\}$ whose facets are described by quadruples $\{a,b,c,d\}$ such that $$b-a=g ({\rm mod}~ n)$$ and $$d-c=h ({\rm mod}~ n)$$.

Let us also consider a variant $L(n;g,h)$ where we further assume that the ordering of $\{a,b,c,d\}$ is cyclic. (Namely, $a<b<c<d$ or $b<c<d<a$ or $c<d<a<b$ or $d<a<b<c$.)

Question 1:

What can be said about the combinatorics and topology of simplicial complexes of the form $K(n;g,h)$ and $L(n;g,h)$.

Question 2:

The construction can be extended to give similar $2k-1$-dimensional simplicial complexes $K(n;g_1,g_2,\dots , g_k)$ and $L(n;g_1,g_2,\dots , g_k)$. What can be said about the combinatorics and topology of these simplicial complexes

Question 3:

Are there nice extensions to odd dimensions?

Motivation

The constructions are motivated by two classes of constructions.

A) When all the $g_1$s are one this is precisely the construction of the cyclic even-dimensional polytopes.

Like them it looks that the complexes considered above (especially the $K's$ have some chance to be pseudomanifolds. Are they ever manifolds? Is there a way to give an arithmetic definition of this kind to notable odd dimensional triangulations like the 6-vertex RP^2 or Kuhnel's CP_2.

B) There is a remarkable simple arithmetic constructions of acyclic complexes by Linial, Meshulam and Rosenthal (see this paper and this post). In dimension 2 you consider all triples $a,b,c$ modulo $n$ ($n$ a prime) such that $a+b+c$ equals one of three numbers $x,y,z$.

Answer 1

This looks like it might resemble the complexes considered in

MR2844711 Catanzaro, Michael J. Generalized Tonnetze. J. Math. Music 5 (2011), no. 2, 117--139.

He considers 2-dimensional simplicial complexes $L(n_1,n_2,n_3)$ with vertex set $\{1,\ldots,N\}$ and $n_1+n_2+n_3 = N$, whose 2-simplices are $\{a, a+n_1, a+n_1+n_2\}$. These parameterize triads of `shape' $(n_1,n_2,n_3)$ in an $N$ note scale, generalizing the diatonic $N=12$ major and minor triads, $(n_1,n_2,n_3) = (4,3,5)$.