There is a natural question from the early days of topology: Given two closed smooth manifolds  and , when does an isomorphism  of integral cohomology rings imply that  and  are homeomorphic or even diffeomorphic? Generally, cohomological rigidity does not hold, such as Poincare homology sphere, three dimensional lens spaces, Milnor's exotic spheres and Donaldson's four-dimensional manifolds. In this talk, I will introduce our recent result about this problem. We prove that, for two classes of manifolds arising from simple polytopes: moment-angle manifolds and toric manifolds (projective toric varieties) are cohomological rigid whenever the polytopes satisfy a combinatorial condition.

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