Invariance of Pontrjagin classes for Bott manifolds

A Bott manifold is the total space of some iterated $\mathbb C P^1$-bundle 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 $\mathbb Z/2$-trivial, where a Bott manifold is called $\mathbb Z/2$-trivial if its cohomology ring with $\mathbb Z/2$-coefficient is isomorphic to that of a product of $\mathbb C P^1$'s.

[1]  Michael Wiemeler,et al.  Torus manifolds and non‐negative curvature , 2014, J. Lond. Math. Soc..

[2]  Suyoung Choi.  Classification of Bott manifolds up to dimension eight , 2011, 1112.2321.

[3]  M. Masuda,et al.  Rigidity problems in toric topology: A survey , 2011, 1102.1359.

[4]  Hiroaki Ishida.  Filtered cohomological rigidity of bott towers , 2010, 1006.4932.

[5]  M. Masuda,et al.  Classification of $\mathbb{Q}$-trivial Bott manifolds , 2009, 0912.5000.

[6]  M. Masuda,et al.  Classification of Q-trivial Bott manifolds , 2009 .

[7]  T. Panov,et al.  Toric cohomological rigidity of simple convex polytopes , 2008, 0807.4800.

[8]  M. Masuda,et al.  Topological classification of generalized Bott towers , 2008, 0807.4334.

[9]  M. Masuda,et al.  Classification problems of toric manifolds via topology , 2007, 0709.4579.

[10]  T. Panov,et al.  Semifree circle actions, Bott towers and quasitoric manifolds , 2006, math/0607094.

[11]  N. Ray,et al.  Homotopy decompositions and K-theory of Bott towers , 2004, math/0408261.

[12]  T. Panov,et al.  On the cohomology of torus manifolds , 2003, math/0306100.

[13]  M. Masuda.  Unitary toric manifolds, multi-fans and equivariant index , 1999 .

[14]  Dennis Sullivan,et al.  Infinitesimal computations in topology , 1977 .

[15]  T. Petrie.  Torus actions on homotopy complex projective spaces , 1973 .

[16]  T. Petrie,et al.  Smooth $S^1 $ actions on homotopy complex projective spaces and related topics , 1972 .

[17]  A. Borel,et al.  CHARACTERISTIC CLASSES AND HOMOGENEOUS SPACES, I.* , 1958 .

[18]  hoi,et al.  Properties of Bott manifolds and cohomological rigidity , 2011 .

[19]  Тарас Евгеньевич Панов,et al.  Полусвободные действия окружности, башни Ботта и квазиторические многообразия@@@Semifree circle actions, Bott towers and quasitoric manifolds , 2008 .

[20]  T E Panov,et al.  Semifree circle actions, Bott towers and quasitoric manifolds , 2008 .

[21]  S. Novikov,et al.  HOMOTOPICALLY EQUIVALENT SMOOTH MANIFOLDS. I , 2004 .