Topological classification of generalized Bott towers

By Suyoung Choi, M. Masuda, D. Suh. 2008

106 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8f211acba8c078147c9b811a26fcb0a365f88efa

How researchers see this paper

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In Discrete Mathematics and Combinatorics

1. M. Masuda
2. D. Suh
3. Suyoung Choi
4. T. Panov
5. Y. Kamishima
6. V. Buchstaber
7. Nigel Ray
8. Frédéric Bosio

In Geometry and Topology

1. M. Masuda
2. D. Suh
3. Suyoung Choi
4. T. Panov
5. Y. Kamishima
6. V. Buchstaber
7. Nigel Ray
8. Frédéric Bosio

In Mathematical Physics

1. M. Masuda
2. T. Panov
3. D. Suh
4. Suyoung Choi

Citations in the corpus

• Toric topology (2012, unknown): “We also have a partial affirmative solution to the cohomological rigidity problem in [13].”
• Toric cohomological rigidity of simple convex polytopes (2008, background): “In [6] it was proved that, if M is a quasitoric manifold over a product of simplices ∏t i=1 Δ ni such that H∗(M) ∼= H∗(∏ti=1 CPi), then M is homeomorphic to ∏ti=1 CPi .”
• Rigidity problems in toric topology: A survey (2011, background): “The variety classification of two-stage generalized Bott manifolds is completed by Kleischmidt [30], and their diffeomorphism classification is done by the authors in [11].”
• Quasitoric manifolds over a product of simplices (2008, background): “We prove in [3] that if a generalized Bott manifold is coh omologically trivial over Z, then it is diffeomorphic to a product of complex projective spaces.”
• Classification problems of toric manifolds via topology (2007, background): “It turns out that the converse holds ([24]), so we have the following another equivalent statement to (V1) above: (V4) H∗T (X) and H ∗ T (Y ) are weakly isomorphic as algebras.”
• Cohomological rigidity of real Bott manifolds (2008, background): “We also prove that the converse is true, namely a real toric manifold which admits a flat riemannian metric invariant under the action of an elementary abelian 2-group is a real Bott manifold.”