Cohomological rigidity of real Bott manifolds (Q1048473)

The term cohomology rigidity refers to a problem from algebraic topology, namely to identity whether the homotopy or homeomorphism type of a topological space is determined by its cohomology ring. There are several variations of this problem; someone can for example ask whether the homotopy type of a give space is determined by it cohomology ring together with the action of the Steenrod algebra. In this paper the authors restrict their attention to toric manifolds asking whether two toric manifolds are diffeomorphic (or homeomorphic) if their cohomology rings with mod 2 coefficients are isomorphic as graded rings. In general there are many closed smooth manifolds which are not homeomorphic but have isomorphic cohomology rings even with integral coefficients. The authors prove that two real Bott manifolds are diffeomorphic if their cohomology rings with \(\mathbb{Z}/2\)-coefficients are isomorphic. In addition the authors prove the converse of a well-known property of real Bott manifolds, that is, 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.