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

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 $n$?

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$j^*$ preserves the degree, so $deg(j^*(x_I))=|I|+deg(x)=|I|+k$.

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    $\begingroup$ 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 $n$? $\endgroup$
    – QSR
    Commented Sep 7, 2015 at 6:12

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