Surprising Examples of Manifolds in Toric Topology!

TL;DR

This paper classifies small covers over certain polytopes in dimensions 4 to 7, verifies the lifting conjecture, and introduces new examples of quasitoric manifolds with high chromatic numbers, advancing understanding in toric topology.

Contribution

It provides a complete classification of small covers for specific polytopes, verifies the lifting conjecture in these cases, and introduces new high-dimensional quasitoric manifolds with complex properties.

Findings

Complete classification of small covers in specified dimensions.

Verification of the lifting conjecture for all considered cases.

First examples of high-dimensional quasitoric manifolds with orbit polytopes of high chromatic number.

Abstract

We investigate small covers and quasitoric over the duals of neighborly simplicial polytopes with small number of vertices in dimensions $4$, $5$, $6$ and $7$. In the most of the considered cases we obtain the complete classification of small covers. The lifting conjecture in all cases is verified to be true. The problem of cohomological rigidity for small covers is also studied and we have found a whole new series of weakly cohomologically rigid simple polytopes. New examples of manifolds provide the first known examples of quasitoric manifolds in higher dimensions whose orbit polytopes have chromatic numbers $\chi (P^n)\geq 3n-5$.