c©2008 ICMS in KAIST COHOMOLOGICAL RIGIDITY OF QUASITORIC MANIFOLDS AND SIMPLE CONVEX POLYTOPES

An n-dimensional convex polytope is called simple if at each vertex exactly n facets (codimension one face) intersect. The notion of quasitoric manifold was first introduced by Davis and Januszkiewicz in [4] as a topological analogue of toric variety in algebraic geometry, which is a closed 2n-dimensional manifold M2n with a locally standard n-torus Tn = (S1)n action such that its orbit space has a combinatorial structure of a simple convex polytope Pn. In this case we say M2n is over Pn. Davis and Januszkiewicz named such manifold as toric manifold, but toric manifold is a well-established term for algebraic geometors as a nonsigular toric variety. So Buchstaber and Panov renamed it as quasitoric manifold in [1], and now this terminology is widely accepted among toric toopolgists. One of the interesting questions on quasitoric manifold is the following cohomo-logical rigidity question. Question 0.1. Let M and N be two quasitoric manifolds with isomorphic coho-mology rings. Does this imply that M and N are homeomorphic? Masuda and Panov give a positive answer to the question for quasitoric mani-folds whose cohomology rings are isomorphic to those of product of CP 1’s in [5]. Namely, they show that any quasitoric manifoldM2n with H∗(M2n) ∼ = H∗(∏CP 1) is actually homeomorphic to CP 1. It is proved in two steps. First they show that any quasitoric manifold M2n over a cube In with H∗(M2n) ∼ = H∗(∏CP 1) is homeomorphic to CP 1. Then they show that any quasitoric manifold M2n with H∗(M2n) ∼ = H∗(∏CP 1) is actually over the cube In