cohomology of iterated loop space on spheres

In the book The homology of iterated loop spaces, 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 $$H^*(\Omega^3S^3;\mathbb{Z}_2)?$$

1 Answer

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).

• Dear Prof, is it possible to write out the explicit cup-product expression of $H^*(\Omega^3S^3;\mathbb{Z}_2)$ under a basis?
– QSR
Sep 2, 2015 at 8:11
• I do not quite understand "The unstable Adams spectral sequence for free iterated loop spaces" Memoirs of the American Mathematical Society, Vol.36 Number 258 (1982). It is quite complicated.
– QSR
Sep 2, 2015 at 8:11
• I didn't say it was easy, I said it had already been worked out. Basically you get the generator in cohomology which are dual to the primitive in homology, and you get the height at which it gets truncated (if it ever does) by looking at the Verschiebung, so you get the ring structure. Anyway, you can read off an "explicit ring structure" of $H^*(\Omega ^3S^3;Z/2)$ from Theorem 3.7, with $X=S^0$, $n=3$. Pairing these elements with the "standard basis" in homology, requires a little bit of work, but it is a routine exercise. Sep 2, 2015 at 9:46
• Don't you have an electronic copy of this book by Wellington? I need it and cannot find it in the web. Aug 31, 2017 at 15:10