Operations on Spaces over Operads and Applications to Homotopy Groups

TL;DR

This paper introduces generalized smash operations on spaces over operads, providing a new algebraic framework to understand the homotopy groups of such spaces, especially double loop spaces, via operadic modules.

Contribution

It develops a conceptual description of homotopy groups over symmetric K(π,1) operads using free algebraic symmetric operads generated by smash operations.

Findings

Homotopy groups form modules over free algebraic symmetric operads.

Double loop space homotopy groups relate to conjugacy classes of Brunnian braids.

Smash operations generalize the Samelson product to spaces over operads.

Abstract

We establish certain smash operations on spaces over operads which are general analogues of the Samelson product on single loop spaces, and obtain a conceptual description of the structure of the homotopy groups of spaces $Y$ over a symmetric $K(\pi,1)$ operad: $\pi_*Y$ is a module over the free algebraic symmetric operad generated by operations on homotopy groups induced by these smash operations. In particular the homotopy groups of double loop spaces is a module over the free algebraic symmetric operad generated by the conjugacy classes of Brunnian braids modulo the conjugation action of pure braids.