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

TL;DR

This paper shows that for generalized real Bott manifolds of height 2, cohomology rings with Z/2 coefficients do not determine their diffeomorphism type, providing a counterexample to the cohomological rigidity problem.

Contribution

It demonstrates the non-rigidity of these manifolds by proving cohomology rings do not distinguish diffeomorphism classes and establishes that homotopy equivalence implies diffeomorphism.

Findings

Cohomology rings with Z/2 coefficients do not distinguish manifolds up to diffeomorphism.

Generalized real Bott manifolds of height 2 are diffeomorphic if homotopy equivalent.

Counterexample to the cohomological rigidity problem for real toric manifolds.

Abstract

We investigate when two generalized real Bott manifolds of height 2 have isomorphic cohomology rings with Z/2 coefficients and also when they are diffeomorphic. It turns out that cohomology rings with Z/2 coefficients do not distinguish those manifolds up to diffeomorphism in general. This gives a counterexample to the cohomological rigidity problem for real toric manifolds posed in \cite{ka-ma08}. We also prove that generalized real Bott manifolds of height 2 are diffeomorphic if they are homotopy equivalent.