Quasitoric manifolds and small covers over properly coloured polytopes: immersions and embeddings (Q2817559)

By the embedding theorem of Whitney every \(n\)-dimensional manifold can be embedded into a euclidean space of dimension \(2n\) and immersed into the \((2n-1)\)-dimensional euclidean space. These bounds on the dimension of the euclidean space are sharp in the sense that for \(n=2^k\) there are manifolds of dimension \(n\) which can not be embedded (or immersed) into a lower-dimensional euclidean space.NEWLINENEWLINEIn this paper the authors construct new examples of manifolds for which these bounds are sharp or almost sharp. These examples are certain small covers and quasitoric manifolds. The proof that the bounds are sharp for these manifolds is an explicit calculation of the dual Stiefel-Whitney classes of these manifolds. These classes serve as obstructions to the existence of embeddings into \(\mathbb{R}^k\).