On Imbedding Differentiable Manifolds in Euclidean Space

Assume n,k,m,q are positive integers. Let M^n denote a smooth differentiable n-manifold and R^k Euclidean k-space.(a) If M^n is open it imbeds smoothly in R^k, k=2n-1(b) If M^n is open and parallelizable it immerses in R^n(c) Assume M^n is closed and (m-1)-connected, 1< 2m-n < n+1. If a neighborhood of the (n-m)-skeleton immerses in R^q,>2n-2m, then the complement of a point of M^n imbeds smoothly in R^q.