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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 pointby iterated pushouts of free $E_k$-algebras.

The May-Milgram filtration and $E_k$-cells