An Additivity Theorem for the Interchange of E_n Structures

TL;DR

This paper proves that the tensor product of cofibrant E_m and E_n operads results in an E_{m+n} operad, clarifying how multiplicative structures combine in higher algebraic topology.

Contribution

It establishes an additivity theorem for E_n operads, showing the tensor product of cofibrant operads yields an operad of higher E-level, resolving a naive conjecture.

Findings

Tensor product of cofibrant E_m and E_n operads is an E_{m+n} operad.

There exists an E_{m_1+m_2+...+m_k} operad mapping into the tensor product of multiple E_{m_i} operads.

Counterexamples to naive conjecture are addressed by cofibrancy conditions.

Abstract

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.