Abstract :

A convex polytope is a convex hull of finitely many points in a Euclidean space. An n-dimensional convex polytope is called simple if at every vertex of the polytope exactly n many facets meet where facets are codimension one faces of the polytope. In this talk, I am going to explain some group actions (mostly by $Z_2^n$ and $ (S^1)^n$ on manifolds such that the orbit space is a simple convex polytope. The concepts of small cover and toric manifold over a simple polytope will be discussed eventually.