Tight and 0-tight polyhedral embeddings of surfaces

In 1] T. Banchoff has studied the problem of high condimensional tight polyhedral embeddings of closed surfaces into Euclidean space. He gave an upper bound for the essential codimension depending on the Euler characteristic of the surface. In 1] and 5] he proved that this bound is attained in some cases and that it is not attained for the Klein bottle. In the present paper we show that this bound is sharp in any case (except the Klein bottle) and that for each surface there exist tight substantial embeddings into Euclidean space of arbitrary dimension up to the Banchoff upper bound. The proof depends essentially on the Heawood map color theorem proved by G. Ringel and J.W.T. Youngs. In addition we get similar results for tight and 0-tight embeddings of surfaces with boundary where it may be remarkable that in case of tight surfaces with one boundary component the Banchoff upper bound can be improved.