An Additivity Theorem for the interchange of En structures

Intuitively one might expect that the tensor product of an Ek-operad with an El-operad (which encode the multiplicative structures of k-fold, respectively l-fold loop spaces) ought to be an Ek+l-operad. However, there are easy counterexamples to this naive conjecture. In this paper we essentially solve the word problem for the nullary, unary, and binary operations of the tensor product of arbitrary topological operads and show that the tensor product of a cofibrant Ek-operad with a cofibrant El-operad is an Ek+l-operad. It follows that if Ai are Eki operads for i=1,2,…,n, then A1⊗…⊗An is at least an Ek1+…+kn operad, i.e. there is an Ek1+…+kn-operad C and a map of operads C→A1⊗…⊗An.