Embedding products of low-dimensional manifolds into Rm

Let X be a Cartesian product of s circles, p orientable 2-manifolds, q non-orientable 2-manifolds, r orientable 3-manifolds and t non-orientable 3-manifolds (all of them are closed). We prove that if either some of these r orientable 3-manifolds embed into R4 or p+q+s+t>0, then the lowest dimension of Euclidean space in which X is smoothly embeddable is s+2p+3(q+r)+4t+1. If none of the closed orientable 3-manifolds R1,…,Rr embed into R4, then their product is embeddable into R3r+2 and, at least for some cases, non-embeddable into R3r+1.