A braided simplicial group

TL;DR

This paper explores the relationship between braid group actions and higher homotopy groups of spheres, revealing new algebraic structures and representations relevant to algebraic topology.

Contribution

It introduces a novel connection between braid group actions and higher homotopy groups of spheres, providing new algebraic insights and spectral sequence representations.

Findings

Higher homotopy group of the 3-sphere as fixed points of braid group action

Representation of higher differentials in Adams spectral sequence

New combinatorial descriptions of related groups

Abstract

By studying braid group actions on Milnor's construction of the 1-sphere, we show that the general higher homotopy group of the 3-sphere is the fixed set of the pure braid group action on certain combinatorially described group. We also give certain representation of higher differentials in the Adams spectral sequence for the homotopy groups of the 2-sphere.