Cohomological non-rigidity of generalized real Bott manifolds of height 2 (Q600723)

The author and \textit{D. Y. Suh} [in: Harada, Megumi (ed.) et al., Toric topology. International conference, Osaka, Japan, May 28--June 3, 2006. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 460, 273--286 (2008; Zbl 1160.57032)] raised the question about whether toric manifolds are diffeomorphic if their cohomology rings with \(\mathbb Z\) coefficients are isomorphic as graded rings. There are partial solutions but the problem is outstanding. Subsequently the author and \textit{Y. Kamishima} [Algebr. Geom. Topol. 9, No. 4, 2479--2502 (2009; Zbl 1195.57071)] asked the same question with regard to real toric manifolds and \(\mathbb Z/2\) cohomology rings. A real Bott manifold is the total space of an iterated \(\mathbb RP^1\) bundle, where each bundle is the projectivization of a Whitney sum of two line bundles. In the Kamishima and Masuda paper already cited and the author's work [Classification of real Bott manifolds; \url{arXiv:0809.2178v2}] the \(\mathbb Z/2\) version is proved for such real Bott manifolds. In the present work the author investigates a generalization to real toric manifolds obtained as the total spaces of the projectivization of Whitney sums of real line bundles over real projective spaces (what is defined as a generalized real Bott manifold of height 2). The \(\mathbb Z/2\) conjecture is shown to fail in general for such real toric manifolds. The author is able to show that if there is homotopy equivalence for these generalized real Bott manifolds of height 2 then there is diffeomorphic equivalence.