Algebraic & Geometric Topology

Invariance of Pontrjagin classes for Bott manifolds

Suyoung Choi, Mikiya Masuda, and Satoshi Murai

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


A Bott manifold is the total space of some iterated 1–bundles over a point. We prove that any graded ring isomorphism between the cohomology rings of two Bott manifolds preserves their Pontrjagin classes. Moreover, we prove that such an isomorphism is induced from a diffeomorphism if the Bott manifolds are 2–trivial, where a Bott manifold is called 2–trivial if its cohomology ring with 2–coefficients is isomorphic to that of a product of copies of 1.

Article information

Algebr. Geom. Topol., Volume 15, Number 2 (2015), 965-986.

Received: 6 May 2014
Revised: 15 September 2014
Accepted: 18 September 2014
First available in Project Euclid: 28 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R19: Algebraic topology on manifolds 57R20: Characteristic classes and numbers

Bott manifold cohomological rigidity Pontrjagin class torus manifold $\Z_2$–trivial Bott manifold


Choi, Suyoung; Masuda, Mikiya; Murai, Satoshi. Invariance of Pontrjagin classes for Bott manifolds. Algebr. Geom. Topol. 15 (2015), no. 2, 965--986. doi:10.2140/agt.2015.15.965.

Export citation


  • A Borel, F Hirzebruch, Characteristic classes and homogeneous spaces, I, Amer. J. Math. 80 (1958) 458–538
  • S Choi, Classification of Bott manifolds up to dimension eight to appear in Proc. Edinburgh Math. Soc.
  • S Choi, M Masuda, Classification of $\mathbb Q$–trivial Bott manifolds, J. Symplectic Geom. 10 (2012) 447–461
  • S Choi, M Masuda, D,Y Suh, Topological classification of generalized Bott towers, Trans. Amer. Math. Soc. 362 (2010) 1097–1112
  • S Choi, M Masuda, D,Y Suh, Rigidity problems in toric topology: A survey, Tr. Mat. Inst. Steklova 275 (2011) 188–201 In Russian; translated in Proc. Stelov Inst. Math. 275 (2011) 177–190
  • S Choi, T,E Panov, D,Y Suh, Toric cohomological rigidity of simple convex polytopes, J. Lond. Math. Soc. 82 (2010) 343–360
  • S Choi, D,Y Suh, Properties of Bott manifolds and cohomological rigidity, Algebr. Geom. Topol. 11 (2011) 1053–1076
  • Y Civan, N Ray, Homotopy decompositions and $K\!$–theory of Bott towers, $K\!$–Theory 34 (2005) 1–33
  • D Crowley, M Kreck, Hirzebruch surfaces, Bulletin of the Manifold Atlas (2011)
  • H Ishida, Filtered cohomological rigidity of Bott towers, Osaka J. Math. 49 (2012) 515–522
  • Manifold atlas project, Petrie conjecture, electronic resource Available at \setbox0\makeatletter\@url {\unhbox0
  • M Masuda, Unitary toric manifolds, multi-fans and equivariant index, Tohoku Math. J. 51 (1999) 237–265
  • M Masuda, T,E Panov, On the cohomology of torus manifolds, Osaka J. Math. 43 (2006) 711–746
  • M Masuda, T,E Panov, Semi-free circle actions, Bott towers, and quasitoric manifolds, Mat. Sb. 199 (2008) 95–122 In Russian; translated in Sb. Math. 199 (2008) 1201–1223
  • M Masuda, D,Y Suh, Classification problems of toric manifolds via topology, from: “Toric topology”, (M Harada, Y Karshon, M Masuda, T,E Panov, editors), Contemp. Math. 460, Amer. Math. Soc. (2008) 273–286
  • T Petrie, Smooth $S\sp{1}$ actions on homotopy complex projective spaces and related topics, Bull. Amer. Math. Soc. 78 (1972) 105–153
  • T Petrie, Torus actions on homotopy complex projective spaces, Invent. Math. 20 (1973) 139–146
  • D Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. (1977) 269–331
  • M Wiemeler, Torus manifolds and non-negative curvature