Ajou University, Korea
Cohomological rigidity problems in Toric topology
As is well-known, cohomology ring does not distinguish closed smooth manifolds up to diffeomorphism or homeomorphism in general. However, it does if we restrict our attention to a reasonably small class of objects. For instance, it is known that simply connected closed smooth 4-manifolds are classified up to homeomorphism using their integral cohomology rings. The classification of the toric manifolds up to homeomorphism or diffeomorphism is an interesting open problem. In particular one can ask whether the integral cohomology ring determines the homeomorphism or diffeomorphism type. So far, we do not have any counter example but affirmative partial results. In this talk, we survey results on the topological classifiation of toric manifolds. If time allows, we also discuss about the topological classification of real toric manifolds.