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The notion of interchange of two multiplicative structures on a topologicalspace is encoded by the tensor product of the two operads parametrizing thesestructures. Intuitively one might thus expect that the tensor product of an E_mand 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 thereare easy counterexamples to this naive conjecture. In this paper we show thatthe tensor product of a cofibrant E_m operad and a cofibrant E_n operad is anE_{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 tensorproduct.

An Additivity Theorem for the Interchange of E_n Structures