Let $n>0$. At the prime $p=2$ the Snaith splitting of $\Omega^kS^{n+k}$ realises the hight/weight filtration on the homology $H_*(\Omega^kS^{n+k};\mathbb{Z}/2)$. If I choose only the pieces $D_{2^r}(\mathbb{R}^k,S^n)$ with $r\geqslant 0$ then homologically, just for the algebra structure, it seems I can recover $H_*(\Omega^kS^{n+k};\mathbb{Z}/2)$ from the free commutative algebra that homology of these pieces generate. Now, the space $\bigvee_{r=0}^{+\infty}D_{2^r}(\mathbb{R}^k,S^n)$ fails to be a monoid under the obvious pairings $D_r\wedge D_s\to D_{r+s}$ as it is not closed really!

I wonder if one can put a multiplicative monoid structure on this wedge so that its group-completion is weak equivalent to $\Omega^kS^{n+k}$.