An Additivity Theorem for the interchange of E-n structures

Abstract

Let A and B be operads and let X be an object with an A-algebra and a B-algebra structure. These structures are said to interchange if each operation alpha : X-n -> X of the A-structure is a homomorphism with respect to the B-structure and vice versa. In this case the combined structure is codified by the tensor product A circle times B of the two operads. There is not much known about A circle times B in general, because the analysis of the tensor product requires the solution of a tricky word problem. Intuitively one might expect that the tensor product of an E-k-operad with an E-l-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 E-k-operad with a cofibrant E-l-operad is an Ek+l-operad. It follows that if A(i) are E-k, operads for i = 1,2,..., n, then A(1) circle times...circle times A(n) is at least an E-kl+...+k(n) operad, i.e. there is an E-k1+...+k(n)-operad C and a map of operads C -> A(1) circle times ... circle times A(n). (C) 2014 Elsevier Inc. All rights reserved.