Generators for unitary bordism ring $\pi_*(MU)$

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)?

• 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. May 4, 2020 at 16:25
• @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? May 4, 2020 at 17:41
• 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 $\pi_4MU$. May 4, 2020 at 17:51
• @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 May 4, 2020 at 19:40