Betti numbers of the geometric spaces associated to nonrational simple convex polytopes

TL;DR

This paper computes the Betti numbers of geometric spaces linked to nonrational simple convex polytopes, revealing they depend on the polytope's combinatorial type just like in the rational case, thus encoding combinatorial features.

Contribution

It extends the understanding of Betti numbers to nonrational polytopes, showing their dependence on combinatorial type similar to rational polytopes.

Findings

Betti numbers depend on combinatorial type in nonrational cases

Betti numbers match those in rational cases for similar polytopes

Combinatorial features are encoded in generalized toric spaces

Abstract

We compute the Betti numbers of the geometric spaces associated to nonrational simple convex polytopes and find that they depend on the combinatorial type of the polytope exactly as in the rational case. This shows that the combinatorial features of the starting polytope are encoded in these generalized toric spaces as they are in their rational counterparts.