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. The algebras constructed this way include many important examples: cohomology algebras of toric varieties and quasitoric manifolds, and Gorenstein algebras of triangulated homology manifolds, introduced and studied 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 nonsingular points are irrelevant. By using bistellar moves we show that the number of distinct dimension vectors arising on a given 3-dimensional pseudomanifold with isolated singularities is a topological invariant. This invariant is trivial on manifolds, but nontrivial on general pseudomanifolds.
This work is supported by the Russian Science Foundation under grant 18-71-00009.
"Dimensions of multi-fan duality algebras." J. Math. Soc. Japan 72 (3) 777 - 794, July, 2020. https://doi.org/10.2969/jmsj/81688168