On quantum topology, hypergraphs and flag vectors

Each rule $f$ that assigns a vector $f(G)$ to an $(n+1)$-graph $G$ determines a class (or property) of $n$-manifold invariants. An invariant $v=v(M)$ is in this class if, for any triangulated manifold $|G|=M$, one has that $v(M)$ is a linear function of $f(G)$. This paper defines a flag vector $f(G)$ for $i$-graphs, whose associated invariants might be quantum, and which is of interest in its own right. The definition (via the concept of shelling, and a `disjoint pair of optional cells' rule for the link) seems to apply to any finite combinatorial object, and so to any compact topological object that can be triangulated. It also applies to finite groups.