<正> In what follows by manifolds and imbeddings we shall mean closed C∞-manifolds and C∞-imbeddings. The notation M~nRN will mean that the n-manifold M~n is imbeddable in the Euclidean Nspace RN. For any integer q>0, let Jq be the set of primes p with 2p≤q+2, and let (Cq,p) stands for the condition Hi(M~n, Zp)=0 for 1≤i≤q. We shall prove: