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We investigate when two generalized real Bott manifolds of height 2 haveisomorphic cohomology rings with Z/2 coefficients and also when they arediffeomorphic. It turns out that cohomology rings with Z/2 coefficients do notdistinguish those manifolds up to diffeomorphism in general. This gives acounterexample to the cohomological rigidity problem for real toric manifoldsposed in \cite{ka-ma08}. We also prove that generalized real Bott manifolds ofheight 2 are diffeomorphic if they are homotopy equivalent.

Cohomological non-rigidity of generalized real Bott manifolds of height 2