The May-Milgram filtration and $E_k$-cells

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 $Ω^m Σ^m S^{k}$ lifts to a filtration of the free $E_{k+m}$-algebra on a point by iterated pushouts of free $E_k$-algebras.

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