Volume 9, issue 4 (2009)

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Cohomological rigidity of real Bott manifolds

Yoshinobu Kamishima and Mikiya Masuda

Algebraic & Geometric Topology 9 (2009) 2479–2502
Abstract

A real Bott manifold is the total space of an iterated ${ℝℙ}^{1}$–bundle over a point, where each ${ℝℙ}^{1}$–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