Problems Concerning Embeddings of Manifolds

Very little is known in general about estimating the smallest integer l such that a manifold Mn emheds in Rn+k +l if it immerses in Rn+k. Indeed there are relatively few examples where k and l can be estimated accurately. There are old examples for which l is known to be arbitrarily Iarge; for those examples l can grow like logn and there are recent examples where l can grow linearly with n. The main difficulty in resolving questions of this kind is that the only general methods known for proving non-embedding and non-immersion results involve doing calculations with characteristic classes and the estimates that they give are very similar for the two problems. In this paper an account is given of various methods that can be used to study examples.