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Cohomological rigidity of real Bott manifolds

Yoshinobu Kamishima and Mikiya Masuda

Algebraic & Geometric Topology 9 (2009) 2479–2502

A real Bott manifold is the total space of an iterated 1–bundle over a point, where each 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 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.

real toric manifold, real Bott tower, flat Riemannian manifold
Mathematical Subject Classification 2000
Primary: 57R91
Secondary: 53C25, 14M25
Received: 28 August 2009
Accepted: 12 October 2009
Published: 13 December 2009
Yoshinobu Kamishima
Department of Mathematics
Tokyo Metropolitan University
Minami-Ohsawa 1-1
Tokyo 192-0397
Mikiya Masuda
Department of Mathematics
Osaka City University
Osaka 558-8585