On the cohomology equivalences between bundle-type quasitoric manifolds over a cube

TL;DR

This paper introduces bundle-type quasitoric manifolds and proves classification results showing cohomology ring isomorphisms imply topological equivalences in specific cases, over a cube and for certain bundles.

Contribution

It establishes the notion of bundle-type quasitoric manifolds and provides classification theorems based on cohomology ring isomorphisms.

Findings

Bundle-type quasitoric manifolds are classified by cohomology rings.

Two ($ ext{CP}^2 ext{#} ext{CP}^2$)-bundle type manifolds are weakly equivariantly homeomorphic if their cohomology rings are isomorphic.

There are only four quasitoric manifolds over $I^3$ not of bundle-type, up to weakly equivariant homeomorphism.

Abstract

The aim of this article is to establish the notion of bundle-type quasitoric manifolds and provide two classification results on them: (1) ($\mathbb{C}P^2\sharp\mathbb{C}P^2$)-bundle type quasitoric manifolds are weakly equivariantly homeomorphic if their cohomology rings are isomorphic, and (2) quasitoric manifolds over $I^3$ are homeomorphic if their cohomology rings are isomorphic. In the latter case, there are only four quasitoric manifolds up to weakly equivariant homeomorphism which are not bundle-type.