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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Topological classification of generalized Bott towers


Authors: Suyoung Choi, Mikiya Masuda and Dong Youp Suh
Journal: Trans. Amer. Math. Soc. 362 (2010), 1097-1112
MSC (2000): Primary 57R19, 57R20, 57S25, 14M25
Published electronically: September 18, 2009
MathSciNet review: 2551516
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Abstract: If $ B$ is a toric manifold and $ E$ is a Whitney sum of complex line bundles over $ B$, then the projectivization $ P(E)$ of $ E$ is again a toric manifold. Starting with $ B$ as a point and repeating this construction, we obtain a sequence of complex projective bundles which we call a generalized Bott tower. We prove that if the top manifold in the tower has the same cohomology ring as a product of complex projective spaces, then every fibration in the tower is trivial so that the top manifold is diffeomorphic to the product of complex projective spaces. This gives supporting evidence to what we call the cohomological rigidity problem for toric manifolds, “ Are toric manifolds diffeomorphic (or homeomorphic) if their cohomology rings are isomorphic?" We provide two more results which support the cohomological rigidity problem.


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Additional Information

Suyoung Choi
Affiliation: Department of Mathematical Sciences, KAIST, 335 Gwahangno, Yuseong-gu, Daejeon 305-701, Republic of Korea
Email: choisy@kaist.ac.kr

Mikiya Masuda
Affiliation: Department of Mathematics, Osaka City University, Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
Email: masuda@sci.osaka-cu.ac.jp

Dong Youp Suh
Affiliation: Department of Mathematical Sciences, KAIST, 335 Gwahangno, Yuseong-gu, Daejeon 305-701, Republic of Korea
Email: dysuh@math.kaist.ac.kr

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04970-8
PII: S 0002-9947(09)04970-8
Keywords: Generalized Bott tower, cohomological rigidity, toric manifold
Received by editor(s): September 16, 2008
Published electronically: September 18, 2009
Additional Notes: The first author was partially supported by the second stage of Brain Korea 21 project, KAIST in 2007, the second author was partially supported by Grant-in-Aid for Scientific Research 4102-17540092, and the third author was partially supported by the SRC program of Korea Science and Engineering Foundation R11-2007-035-02002-0.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.