# dimension of generators of cohomology ring of iterated loop-suspension

In the book The unstable Adams spectral sequence for free iterated loop spaces, R.J. Wellington, Mem. Amer. Math. Soc. 258, 1982, p. 32

Question: When $p=2$, $k\geq 1$, $n=0$ to $\infty$, what kind of $I$ can we choose? How is the exterior algebra $H^*(\Omega^{n+1}\Omega^{n+k+1};\mathbb{Z}_2)$ related to $n$?

The related context is in the following.

$j^*$ preserves the degree, so $deg(j^*(x_I))=|I|+deg(x)=|I|+k$.
• Thanks, Prof. I have another question: When $p=2$, $k\geq 1$, $n=0$ to $\infty$, what kind of $I$ can we choose? How is the exterior algebra $H^*(\Omega^{n+1}\Omega^{n+k+1};\mathbb{Z}_2)$ related to $n$? – QSR Sep 7 '15 at 6:12