Cohomological rigidity of real Bott manifolds

Kamishima, Yoshinobu; Masuda, Mikiya

doi:10.2140/agt.2009.9.2479

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Cohomological rigidity of real Bott manifolds

Yoshinobu Kamishima and Mikiya Masuda

Algebraic & Geometric Topology 9 (2009) 2479–2502

DOI: 10.2140/agt.2009.9.2479

Abstract

A real Bott manifold is the total space of an iterated ${ℝ ℙ}^{1}$ –bundle over a point, where each ${ℝ ℙ}^{1}$ –bundle is the projectivization of a Whitney sum of two real line bundles. We prove that two real Bott manifolds are diffeomorphic if their cohomology rings with $ℤ ∕ 2$ –coefficients are isomorphic.

A real Bott manifold is a real toric manifold and admits a flat Riemannian metric invariant under the natural action of an elementary abelian 2–group. We also prove that the converse is true, namely a real toric manifold which admits a flat Riemannian metric invariant under the action of an elementary abelian 2–group is a real Bott manifold.

Keywords

real toric manifold, real Bott tower, flat Riemannian manifold

Mathematical Subject Classification 2000

Primary: 57R91

Secondary: 53C25, 14M25

References

Publication

Received: 28 August 2009

Accepted: 12 October 2009

Published: 13 December 2009

Authors

Yoshinobu Kamishima
Department of Mathematics Tokyo Metropolitan University Minami-Ohsawa 1-1 Hachioji Tokyo 192-0397 Japan

Mikiya Masuda
Department of Mathematics Osaka City University Sumiyoshi-ku Osaka 558-8585 Japan

Volume 9, issue 4 (2009)