Permutohedral structures on \(E_2\)-operads. (English) Zbl 1454.55010
Summary: There are two interesting families of \(E_{2}\)-operads, those that detect double loop spaces, and those that solve Deligne’s conjecture on Hochschild cochains. The first family deformation retracts to Milgram’s model obtained by gluing together permutohedra along their faces. We show how the second family can be covered by permutohedra as well, shedding new light on several proposed solutions of Deligne’s conjecture. In particular, our approach induces an explicit homotopy equivalence between the models of the two families. The permutohedra and partial orders play a central role providing direct links to other fields of mathematics. We for instance find a new cellular decomposition of permutohedra using partial orders and that the permutohedra give the cells for the Dyer-Lashof operations.
MSC:
| 55P48 | Loop space machines and operads in algebraic topology |
| 55P35 | Loop spaces |
| 55R80 | Discriminantal varieties and configuration spaces in algebraic topology |
| 52B12 | Special polytopes (linear programming, centrally symmetric, etc.) |
Online Encyclopedia of Integer Sequences:
Coefficients of list partition transform: reciprocal of an exponential generating function (e.g.f.).References:
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