Properties of Bott manifolds and cohomological rigidity

Details Properties of Bott manifolds and cohomological rigidity Properties of Bott manifolds and cohomological rigidity

2009-12-16

arxiv.org/abs/0912.3054v1

Algebr. Geom. Topol. 11(2), (2011), 1053--1076

Mathematics

Algebraic Topology

22 pages

Scientific paper

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.

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