HR-length of a free group via polynomial functors

TL;DR

This paper establishes a lower bound on the length of the Bousfield $HR$-localization tower for free groups of rank at least 2, using polynomial functors over $Q$ to advance understanding in algebraic topology.

Contribution

It introduces a novel application of polynomial functors over $Q$ to analyze the $HR$-localization tower of free groups, providing new lower bounds.

Findings

The $HR$-localization tower length is at least $oldsymbol{ ext{omega}+ ext{omega}}$ for free groups of rank ≥ 2.

Polynomial functors over $Q$ are key tools in the proof.

The result advances the understanding of localization towers in algebraic topology.

Abstract

We prove that for a subring $R\subseteq \mathbb Q$ and a free group $F$ of rank at least $2$ the length of the Bousfield's $HR$-localization tower for $F$ is at least $\omega+\omega$. The key ingredient of the proof is the theory of polynomial functors over $\mathbb Q.$