arXiv subjects
Publications and source records attributed to Jonathan Fine.
This note defines a flag vector for $i$-graphs. The construction applies to any finite combinatorial object that can be shelled. Two possible connections to quantum topology are mentioned. Further details appear in the author's "On quantum topology, hypergraphs and flag vectors", (preprint q-alg/9708001).
This paper defines new intersection homology groups. The basic idea is this. Ordinary homology is locally trivial. Intersection homology is not. It may have significant local cycles. A local-global cycle is defined to be a family of such local cycles that is, at the same time, a global cycle. The motivating problem is the numerical characterisation of the flag vectors of convex polytopes. Central is a study of the cycles on a cone and a cylinder, in terms of those on the base. This leads to the topological definition of local-global intersection homology, and a formula for the expected Betti numbers of toric varieties. Various related questions are also discussed.
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.