On injectivity of Cohen-Wu homomorphism

TL;DR

This paper provides a new elementary, group-theoretic proof of the injectivity of a natural map from Milnor's construction to the pure braid simplicial group, avoiding Lie algebra methods.

Contribution

It introduces a novel, elementary proof of injectivity that is purely group-theoretic, differing from previous Lie algebra-based approaches.

Findings

Proves injectivity of the map from Milnor's construction to pure braid group

Uses elementary, group-theoretic methods rather than Lie algebra techniques

Simplifies understanding of the homomorphism's properties

Abstract

We give a new elementary proof of the theorem that a natural map from Milnor's construction $F[S^1]$ to the simplicial group $\mathrm{AP}$ of pure braids is injective. Our approach is group-theoretic and does not rely on Lie algebras.