I've been trying to understand the calculation of $H_*(\Omega^2 S^{n+2},\mathbb Z/2)$ as an algebra, together with the dual Steenrod operations on it which lower the degrees. I've spent many hours looking for references, and it seems to me that the only place I can find such a computation is in The Homology of Iterated Loop Spaces by May, Cohen and Lada. The thing is that their computation relies on very technical machinery like Dyer Lashof operations and Nishida relations, and they actually show way more than what I just need. Is there any place where I can find a more accessible reference, or should I try to rewrite the whole computation of Cohen translated for my particular case ?
Thank you very much
I went into the details and eventually realised that the computation is made really easy by the following theorem of Borel :
Let $\Omega B\to PB\to B$ be the path fibration of a simply connected $H$-space $B$. Let $f_1,f_2,\dots\in H_*(\Omega B;\mathbf F_2)$ be a locally finite familly of elements such that $\sigma(f_1), \sigma(f_2),\dots$ form a simple system of generators of $H_*(B;\mathbf F_2)$. Then, $$H_*(\Omega B;\mathbf F_2)\cong \mathbf F_2[f_1,f_2,\dots]$$ ((Switzer, theorem 15.60))
It applies in this exact version, with $B=\Omega S^{n+2}$ and $f_i=Q(Q(Q(\cdots Q(x_0))))$ $i$-times, where $x_0$ is the fundamental class of $S^n$ and $Q$ is the only Dyer Lashof operation