On the classification of quasitoric manifolds over the dual cyclic polytopes

TL;DR

This paper classifies quasitoric manifolds and small covers over dual cyclic polytopes, providing a topological understanding of these manifolds for specific dimensions and polytope configurations.

Contribution

It offers a topological classification of quasitoric manifolds over dual cyclic polytopes for certain dimensions and classifies small covers over all such polytopes.

Findings

Classification of quasitoric manifolds over dual cyclic polytopes for n>3 or m-n=3

Complete classification of small covers over all dual cyclic polytopes

Advances understanding of topological structures of these manifolds

Abstract

For a simple $n$-polytope $P$, a quasitoric manifold over $P$ is a $2n$-dimensional smooth manifold with a locally standard action of the $n$-dimensional torus for which the orbit space is identified with $P$. This paper shows the topological classification of quasitoric manifolds over the dual cyclic polytope $C^n(m)^*$, when $n>3$ or $m-n=3$. Besides, we classify small covers, the "real version" of quasitoric manifolds, over all dual cyclic polytopes.