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I’m reading Pengelley’s paper “The mod 2 homology of $MSO$ and $MSU$ as $\mathfrak A^*$ comodule algebras, and the cobordism ring”.

He has chosen very special generators $z_n \in H_n(MO; \mathbb F_2)$.

If $n = 2^t-1$, then $z_{2^t-1} = \varphi(w_{2^t-1}^*)$, where $\varphi$ is the Thom isomorphism. If $n \neq 2^t -1$, then $z_n$ is the Hurewicz image of a $n$-manifold $M_n$. Here $\{M_n\}_{n \neq 2^t-1}$ is a set of polynomial generators of $\pi_*(MO)$ with the additional properties that $M_{2k-1}$ is an orientable submanifold of $M_{2k}$ dual to $w_1(M_{2k})$ which is itself the reduction of an integral class.

These generators are useful to explicitly describe $\mathfrak A_2^*$ comodule structure of $H_*(MSO;\mathbb F_2)$ using the structure of $H_*(MO; \mathbb F_2)$.

Here is the question. Is there a similar set of polynomial generators $\{M_n\in \pi_{2n}(MU)\}_{n\neq p^t -1}$ ($p$ is odd) with the property that $M_{2k-1}$ is dual to $c_1(M_{2k})$? Or is there another way to describe the $\mathfrak A_p$-comodule structure of $H_*(MSU; \mathbb F_p)$ ($p$ is odd)?

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  • $\begingroup$ Are the $M_n$ supposed to be polynomial generators of $\pi_*MU$? If so, it seems like the degree restriction $n\neq p^t-1$ will cause problems. $\endgroup$ May 4 '20 at 16:25
  • $\begingroup$ @JohnGreenwood yes, $M_n$ are supposed to be polynomial generators of $\pi_*(MU)$, but $M_n$ is $2n$-dimensional, so they should be polynomial generators of degree $2n \neq 2(p^t-1)$. What kind of problems could appear there? $\endgroup$ May 4 '20 at 17:41
  • $\begingroup$ Well if $p=3$ for example then you're only allowing polynomial generators of degree 2,6,8,10.... Since you're missing a generator of degree 4, you'll miss one of the two copies of $\mathbb{Z}$ in $\pi_4MU$. $\endgroup$ May 4 '20 at 17:51
  • $\begingroup$ @JohnGreenwood ok, I see. But the question is not about taking polynomial generators for whole $\pi_*(MU)$. I just want to take some of them (with the required property) and after that take their Hurewicz images as the missing part of polynomial generators of $H_*(MU)$. Actually, I'm wondering if there a nice description of $H_*(MSU; \mathbb F_p)$ as an $\mathfrak A_p$ comodule $\endgroup$ May 4 '20 at 19:40

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