Quasitoric manifolds and small covers over properly coloured polytopes: Immersions and embeddings

Abstract

We construct small covers and quasitoric manifolds over n-dimensional simple polytopes which allow proper colourings of facets with n colours. We calculate the Stiefel-Whitney classes of these manifolds as obstructions to immersions and embeddings into Euclidean spaces. The largest dimension required for embedding is achieved in the case n is a power of two.