There is a natural question from the early days of differential topology: Given two closed smooth manifolds M and M', when does an isomorphism of their integral cohomology rings imply that M and M' are homeomorphic or even diffeomorphic? Generally, cohomological rigidity can not hold, such as the Poincare 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 rigidity theorems 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.
