In the book [The homology of iterated loop spaces](http://www.math.uchicago.edu/~may/BOOKS/homo_iter.pdf), the homology Hopf algebra


(1)
$$
H\_\*(\Omega^n \Sigma^n X;\mathbb{Z}\_p)
$$ 


for primes $p\geq 2$ is obtained on p. 226, Thm. 3.2. In particular, the homology Hopf algebra


(2)
$$
H\_\*(\Omega^nS^n;\mathbb{Z}\_2)
$$
is known. However, when I use the Cartan formula and Adem relations to compute the coproduct of (2), I find it is quite complicated when modulo the Adem relations, and do not know how to get the explicit expression of the coproduct. 


Question: Are there any references where I can find the explicit expression of the cohomology algebra
$$
H^\*(\Omega^nS^n;\mathbb{Z}\_2)?
$$


I obtain that $\Omega S^1=\text{Map}\_\*(S^1,S^1)\simeq [S^1;S^1]\_\*=\pi\_1(S^1)=\mathbb{Z}$. Hence $H^\*(\Omega S^1;\mathbb{Z}\_2)=\oplus\_{k\in\mathbb{Z}} \mathbb{Z}\_2a\_k$, $|a\_k|=0$. How to compute the following examples


$$
H^\*(\Omega^2S^2;\mathbb{Z}\_2)
$$
and
$$
H^\*(\Omega^3S^3;\mathbb{Z}\_2)?
$$



cohomology of iterated loop space on spheres
As is known from J.W. Milnor, J.C. Moore, "On the structure of Hopf algebras" Ann. of Math. (2), 81 : 2 (1965) pp. 211–264, the algebra structure of an underlying algebra of a Hopf algebra is quite limited, so that it suffices to study $p$-th powers to determine the algebra structure. Dually, it suffices to 
study the Verschiebung, and not all the coproduct. This has been worked out by 
Wellington, in "The unstable Adams spectral sequence for free iterated loop spaces" Memoirs of the American Mathematical Society, Vol.36 Number 258 (1982).


