Toric origami structures on quasitoric manifolds

TL;DR

This paper constructs higher-dimensional quasitoric manifolds that cannot be realized as toric origami manifolds, linking topological and combinatorial properties.

Contribution

It introduces a method to distinguish quasitoric manifolds from toric origami manifolds using combinatorial and topological criteria.

Findings

Existence of quasitoric manifolds not equivalent to any toric origami manifold in dimensions 6 and higher

Reformulation of the problem in terms of planar triangulations with specific properties

Development of topological and combinatorial tools for classification

Abstract

We construct quasitoric manifolds of dimension 6 and higher which are not equivariantly homeomorphic to any toric origami manifold. All necessary topological definitions and combinatorial constructions are given and the statement is reformulated in discrete geometrical terms. The problem reduces to existence of planar triangulations with certain coloring and metric properties.