Topological classification of generalized Bott towers
Authors:
Suyoung Choi, Mikiya Masuda and Dong Youp Suh
Journal:
Trans. Amer. Math. Soc. 362 (2010), 10971112
MSC (2000):
Primary 57R19, 57R20, 57S25, 14M25
Published electronically:
September 18, 2009
MathSciNet review:
2551516
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: If is a toric manifold and is a Whitney sum of complex line bundles over , then the projectivization of is again a toric manifold. Starting with 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.
 1.
A.
Borel and F.
Hirzebruch, Characteristic classes and homogeneous spaces. I,
Amer. J. Math. 80 (1958), 458–538. MR 0102800
(21 #1586)
 2.
S. Choi, T. Panov and D. Y. Suh,
Toric cohomological rigidity of simple convex polytopes, arXiv:0807.4800.
 3.
S. Choi, M. Masuda and D. Y. Suh,
Quasitoric manifolds over a product of simplices, Osaka J. Math. (to appear), arXiv:0803.2749.
 4.
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
(96i:22030), 10.1215/S0012709494076023
 5.
P.
E. Jupp, Classification of certain 6manifolds, Proc.
Cambridge Philos. Soc. 73 (1973), 293–300. MR 0314074
(47 #2626)
 6.
Peter
Kleinschmidt, A classification of toric varieties with few
generators, Aequationes Math. 35 (1988),
no. 23, 254–266. MR 954243
(89f:14056), 10.1007/BF01830946
 7.
Mikiya
Masuda, Equivariant cohomology distinguishes toric manifolds,
Adv. Math. 218 (2008), no. 6, 2005–2012. MR 2431667
(2009j:14067), 10.1016/j.aim.2008.04.002
 8.
M.
Masuda and T.
E. Panov, Semifree 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. 78, 1201–1223 (2008). MR 2452268
(2009i:57073), 10.1070/SM2008v199n08ABEH003959
 9.
M. Masuda and D. Y. Suh,
Classification problems of toric manifolds via topology, Proc. of Toric Topology, Contemp. Math. 460: 273286, 2008, arXiv:0709.4579.
 10.
John
W. Milnor and James
D. Stasheff, Characteristic classes, Princeton University
Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. Annals of
Mathematics Studies, No. 76. MR 0440554
(55 #13428)
 11.
Franklin
P. Peterson, Some remarks on Chern classes, Ann. of Math. (2)
69 (1959), 414–420. MR 0102807
(21 #1593)
 12.
C.
T. C. Wall, Classification problems in differential topology. V. On
certain 6manifolds, Invent. Math. 1 (1966), 355374; corrigendum,
ibid 2 (1966), 306. MR 0215313
(35 #6154)
 1.
 A. Borel and F. Hirzebruch,
Characteristic classes and homogeneous spaces I, Amer. J. Math., 80: 458538, 1958. MR 0102800 (21:1586)
 2.
 S. Choi, T. Panov and D. Y. Suh,
Toric cohomological rigidity of simple convex polytopes, arXiv:0807.4800.
 3.
 S. Choi, M. Masuda and D. Y. Suh,
Quasitoric manifolds over a product of simplices, Osaka J. Math. (to appear), arXiv:0803.2749.
 4.
 M. Grossberg and Y. Karshon,
Bott towers, complete integrability, and the extended character of representations, Duke Math. J., 76: 2358, 1994. MR 1301185 (96i:22030)
 5.
 P. E. Jupp,
Classification of certain manifolds, Proc. Cambridge Philos. Soc., 73: 293300, 1973. MR 0314074 (47:2626)
 6.
 P. Kleinschmidt,
A classification of toric varieties with few generators, Aequationes Math. 35: 254266, 1988. MR 954243 (89f:14056)
 7.
 M. Masuda,
Equivariant cohomology distinguishes toric manifolds, Adv. Math. 218: 20052012, 2008. MR 2431667
 8.
 M. Masuda and T. E. Panov,
Semifree circle actions, Bott towers, and quasitoric manifolds, Sbornik Math. 199:8, 12011223, 2008. MR 2452268
 9.
 M. Masuda and D. Y. Suh,
Classification problems of toric manifolds via topology, Proc. of Toric Topology, Contemp. Math. 460: 273286, 2008, arXiv:0709.4579.
 10.
 J. W. Milnor and J. D. Stasheff,
Characteristic Classes, Ann. of Math. Studies 76, Princeton N.J. 1974. MR 0440554 (55:13428)
 11.
 F. P. Peterson,
Some remarks on Chern classes, Ann. of Math. (2), 69: 414420, 1959. MR 0102807 (21:1593)
 12.
 C. T. C. Wall,
Classification problems in differential topology. V. On certain manifolds Invent. Math., 1: 355374, 1966. MR 0215313 (35:6154)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2000):
57R19,
57R20,
57S25,
14M25
Retrieve articles in all journals
with MSC (2000):
57R19,
57R20,
57S25,
14M25
Additional Information
Suyoung Choi
Affiliation:
Department of Mathematical Sciences, KAIST, 335 Gwahangno, Yuseonggu, Daejeon 305701, Republic of Korea
Email:
choisy@kaist.ac.kr
Mikiya Masuda
Affiliation:
Department of Mathematics, Osaka City University, Sugimoto, Sumiyoshiku, Osaka 5588585, Japan
Email:
masuda@sci.osakacu.ac.jp
Dong Youp Suh
Affiliation:
Department of Mathematical Sciences, KAIST, 335 Gwahangno, Yuseonggu, Daejeon 305701, Republic of Korea
Email:
dysuh@math.kaist.ac.kr
DOI:
http://dx.doi.org/10.1090/S0002994709049708
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 GrantinAid for Scientific Research 410217540092, and the third author was partially supported by the SRC program of Korea Science and Engineering Foundation R112007035020020.
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
