Dimensions of multi-fan algebras

TL;DR

This paper investigates the dimensions of multi-fan algebras associated with simplicial cycles, disproves a conjecture about their invariance, and introduces a topological invariant related to singularities.

Contribution

It demonstrates that the dimensions of these algebras depend on singular point colors, disproving a previous conjecture, and establishes a topological invariant for 3D pseudomanifolds.

Findings

Dimensions depend on singular point colors

Disproved the conjecture of invariance for all simplicial cycles

Introduced a topological invariant for 3D pseudomanifolds

Abstract

Given an arbitrary non-zero simplicial cycle and a generic vector coloring of its vertices, there is a way to produce a graded Poincare duality algebra associated with these data. The procedure relies on the theory of volume polynomials and multi-fans. This construction includes many important examples, such as cohomology of toric varieties and quasitoric manifolds, and Gorenstein algebras of triangulated homology manifolds, introduced by Novik and Swartz. In all these examples the dimensions of graded components of such duality algebras do not depend on the vector coloring. It was conjectured that the same holds for any simplicial cycle. We disprove this conjecture by showing that the colors of singular points of the cycle may affect the dimensions. However, the colors of smooth points are irrelevant. By using bistellar moves we show that the number of different dimension vectors arising on a given 3-dimensional pseudomanifold with isolated singularities is a topological invariant. This invariant is trivial on manifolds, but nontrivial in general.