Topological classification of generalized Bott towers

Choi, Suyoung; Masuda, Mikiya; Suh, Dong

doi:10.1090/S0002-9947-09-04970-8

Topological classification of generalized Bott towers
HTML articles powered by AMS MathViewer

by Suyoung Choi, Mikiya Masuda and Dong Youp Suh PDF

Trans. Amer. Math. Soc. 362 (2010), 1097-1112 Request permission

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.

References

A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces. I, Amer. J. Math. 80 (1958), 458–538. MR 102800, DOI 10.2307/2372795
S. Choi, T. Panov and D. Y. Suh, Toric cohomological rigidity of simple convex polytopes, arXiv:0807.4800.
S. Choi, M. Masuda and D. Y. Suh, Quasitoric manifolds over a product of simplices, Osaka J. Math. (to appear), arXiv:0803.2749.
Michael Grossberg and Yael Karshon, Bott towers, complete integrability, and the extended character of representations, Duke Math. J. 76 (1994), no. 1, 23–58. MR 1301185, DOI 10.1215/S0012-7094-94-07602-3
P. E. Jupp, Classification of certain $6$-manifolds, Proc. Cambridge Philos. Soc. 73 (1973), 293–300. MR 314074, DOI 10.1017/s0305004100076854
Peter Kleinschmidt, A classification of toric varieties with few generators, Aequationes Math. 35 (1988), no. 2-3, 254–266. MR 954243, DOI 10.1007/BF01830946
Mikiya Masuda, Equivariant cohomology distinguishes toric manifolds, Adv. Math. 218 (2008), no. 6, 2005–2012. MR 2431667, DOI 10.1016/j.aim.2008.04.002
M. Masuda and T. E. Panov, Semi-free circle actions, Bott towers, and quasitoric manifolds, Mat. Sb. 199 (2008), no. 8, 95–122 (Russian, with Russian summary); English transl., Sb. Math. 199 (2008), no. 7-8, 1201–1223 (2008). MR 2452268, DOI 10.1070/SM2008v199n08ABEH003959
M. Masuda and D. Y. Suh, Classification problems of toric manifolds via topology, Proc. of Toric Topology, Contemp. Math. 460: 273–286, 2008, arXiv:0709.4579.
John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554
Franklin P. Peterson, Some remarks on Chern classes, Ann. of Math. (2) 69 (1959), 414–420. MR 102807, DOI 10.2307/1970191
C. T. C. Wall, Classification problems in differential topology. V. On certain $6$-manifolds, Invent. Math. 1 (1966), 355–374; corrigendum, ibid. 2 (1966), 306. MR 215313, DOI 10.1007/BF01425407