An Additivity Theorem for the Interchange of E_n Structures

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.