The May-Milgram filtration and $E_k$-cells

TL;DR

This paper explores the structure of free $E_{k+1}$-algebras and their filtrations, revealing how May-Milgram filtrations lift to $E_{k+m}$-algebras through iterated pushouts, advancing understanding of algebraic topology.

Contribution

It introduces an $E_k$-cell structure on free $E_{k+1}$-algebras and details how May-Milgram filtrations lift to higher $E_{k+m}$-algebras via iterated pushouts.

Findings

Established an $E_k$-cell structure on free $E_{k+1}$-algebras.

Described the lifting of May-Milgram filtrations to $E_{k+m}$-algebras.

Connected filtrations with iterated pushouts of free $E_k$-algebras.

Abstract

We describe an $E_k$-cell structure on the free $E_{k+1}$-algebra on a point, and more generally describe how the May-Milgram filtration of $\Omega^m \Sigma^m S^{k}$ lifts to a filtration of the free $E_{k+m}$-algebra on a point by iterated pushouts of free $E_k$-algebras.