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dimension of generators of cohomology ring of infiniteiterated 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. 32enter image description here

Question: When $p=2$, $k\geq 2$ and$k\geq 1$, $n=\infty$$n=0$ to $\infty$, what is the dimensionkind of each generator $j^*(x_I^*)$$I$ can we choose? I do not quite understandHow is the argument.exterior algebra $H^*(\Omega^{n+1}\Omega^{n+k+1};\mathbb{Z}_2)$ related to $n$?

The related context is in the following.

enter image description here

enter image description here

dimension of generators of cohomology ring of infinite 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. 32enter image description here

Question: When $p=2$, $k\geq 2$ and $n=\infty$, what is the dimension of each generator $j^*(x_I^*)$? I do not quite understand the argument.

The related context is in the following.

enter image description here

enter image description here

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. 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 $n$?

The related context is in the following.

enter image description here

enter image description here

Source Link
QSR
  • 2.2k
  • 10
  • 26

dimension of generators of cohomology ring of infinite 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. 32enter image description here

Question: When $p=2$, $k\geq 2$ and $n=\infty$, what is the dimension of each generator $j^*(x_I^*)$? I do not quite understand the argument.

The related context is in the following.

enter image description here

enter image description here