  The notion of interchange of two multiplicative structures on a topological
space is encoded by the tensor product of the two operads parametrizing these
structures. Intuitively one might thus expect that the tensor product of an E_m
and an E_n operad (which encode the muliplicative structures of m-fold,
respectively n-fold loop spaces) ought to be an E_{m+n} operad. However there
are easy counterexamples to this naive conjecture. In this paper we show that
the tensor product of a cofibrant E_m operad and a cofibrant E_n operad is an
E_{m+n} operad. It follows that if A_i are E_{m_i} operads for i=1,2,...,k,
then there is an E_{m_1+m_2+...+m_k} operad which maps into their tensor
product.

Algebraic Topology