Rigidity of embeddings of finite products of certain continua

We show that an embedding X1×…×Xn→Y1×…×Yn of products k-dimensional continua is factorwise rigid provided each Xi contains a dense family of “cohomological holes” and each Yj is cohomologically (k−1)-connected. As a corollary we obtain that any embedding of Mkm×…×Mkm into itself, where Mkm are the Menger spaces with 1≤k