Properties of Bott manifolds and cohomological rigidity

TL;DR

This paper proves that for certain Bott and quasitoric manifolds, the cohomology ring uniquely determines their topological type, establishing cohomological rigidity in these classes.

Contribution

It demonstrates cohomological rigidity for one-twist Bott manifolds and extends the result to quasitoric manifolds, introducing the concept of twist number and cohomological complexity.

Findings

Cohomology ring determines the topological type of one-twist Bott manifolds.

The twist number equals the cohomological complexity.

Cohomology Bott manifolds are homeomorphic to Bott manifolds.

Abstract

The cohomological rigidity problem for toric manifolds asks whether the cohomology ring of a toric manifold determines the topological type of the manifold. In this paper, we consider the problem with the class of one-twist Bott manifolds to get an affirmative answer to the problem. We also generalize the result to quasitoric manifolds. In doing so, we show that the twist number of a Bott manifold is well-defined and is equal to the cohomological complexity of the cohomology ring of the manifold. We also show that any cohomology Bott manifold is homeomorphic to a Bott manifold. All these results are also generalized to the case with $\mathbb Z_{(2)}$-coefficients, where $\mathbb Z_{(2)}$ is the localized ring at 2.