Cohomological non-rigidity of eight-dimensional complex projective towers (Q2346720)

In this paper the authors consider iterated projectivized complex vector bundles, starting from a point. They call such iterated bundles \({\mathbb C} P\) towers, and their total spaces \({\mathbb C} P\) manifolds. In a previous paper [Proc. Steklov Inst. Math. 286, 285--307 (2014) and Tr. Mat. Inst. Steklova 286, 308--330 (2014; Zbl 1373.57048)] the authors proved that the diffeomorphism type of six-dimensional \({\mathbb C} P\) manifolds are determined by their integer cohomology rings. In the paper at hand they show that this statement is not true in dimension \(8\), by considering projectivizations of complex rank \(2\) vector bundles over \({\mathbb C} P^3\). The total spaces of these bundles are relevant for the problem because their cohomology rings are determined by the first and second Chern classes of the vector bundle, but their homotopy type is not.