Permutahedral Structures of $E_2$ Operads

TL;DR

This paper demonstrates that certain $E_2$ operads can be described using permutahedra, providing new topological insights and explicit cellular decompositions relevant to Deligne's conjecture and related algebraic structures.

Contribution

It shows how $E_2$ operads related to Deligne's conjecture can be covered by permutahedra, establishing a homotopy equivalence and offering a new topological and combinatorial perspective.

Findings

Permutahedra can cover $E_2$ operads related to Deligne's conjecture.

The quotient of permutahedra forms an operad up to homotopy.

A new cellular decomposition of permutahedra using partial orders.

Abstract

There are basically two interesting breeds of $E_2$ operads, those that detect loop spaces and those that solve Deligne's conjecture. The former deformation retract to Milgram's space obtained by gluing together permutahedra at their faces. We show how the second breed can be covered by permutahedra as well. Even more is true, the quotient is actually already an operad up to homotopy, which induces the operad structure on cellular chains adapted to prove Deligne's conjecture, while no such structure is known on Milgram's space. We show, explicitely, that these two quotients are homotopy equivalent. This gives a new topological proof that operads of this type are indeed of the right homotopy type. It also furnishes a very nice clean description in terms of polyhedra, and with it PL topology, for the whole story. The permutahedra and partial orders play a central role. This, in turn, provides direct links to other fields of mathematics. We for instance find a new cellular decomposition of permutahedra using partial orders and that the permutahedra give the cells for the Dyer--Lashof operations.