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Properties of Bott manifolds and cohomological rigidity

Suyoung Choi and Dong Youp Suh

Algebraic & Geometric Topology 11 (2011) 1053–1076

The cohomological rigidity problem for toric manifolds asks whether the integral 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 (2)–coefficients, where (2) is the localized ring at 2.

toric manifold, quasitoric manifold, Bott tower, twist number, cohomological complexity, cohomological rigidity, one-twisted Bott tower
Mathematical Subject Classification 2000
Primary: 57S25
Secondary: 22F30
Received: 24 April 2010
Revised: 31 October 2010
Accepted: 3 January 2011
Published: 30 March 2011
Suyoung Choi
Department of Mathematics
Ajou University
San 5
Suwon 443-749
Republic of Korea
Dong Youp Suh
Department of Mathematical Sciences
335 Gwahangno
Yu-sung Gu
Daejeon 305-701
Republic of Korea