Invariance of Pontrjagin classes for Bott manifolds

Choi, Suyoung¹ ; Masuda, Mikiya² ; Murai, Satoshi³

¹Department of Mathematics, Ajou University, San 5, Woncheon-dong, Yeongtong-gu, Suwon 443-749, South Korea
²Department of Mathematics, Osaka City University, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka-shi 558-8585, Japan
³Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan

Algebraic and Geometric Topology, Tome 15 (2015) no. 2, pp. 965-986

Résumé

A Bott manifold is the total space of some iterated ℂℙ1–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 ℤ∕2–trivial, where a Bott manifold is called ℤ∕2–trivial if its cohomology ring with ℤ∕2–coefficients is isomorphic to that of a product of copies of ℂℙ1.

Détail

DOI : 10.2140/agt.2015.15.965

Classification : 57R19, 57R20
Keywords: Bott manifold, cohomological rigidity, Pontrjagin class, torus manifold, $\mathbb{Z}_2$–trivial Bott manifold

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