# 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.

## Key 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.