I am trying to proof Novikov theorem

\begin{equation} MSU_*\otimes \mathbb Z[\frac 1 2] \cong \mathbb Z[\frac 1 2][y_2, y_4, \ldots],\quad \deg y_i = 2i. \end{equation}

This can be proved by using Adams Spectral Sequence (ASS) for $p \neq 2$, but to use ASS we need to know structure of $H_*(MSU; \mathbb F_p)$ as a module over dual to Steenrod algebra $\mathfrak A_p^*$.

So my question is how to do it?

I seems that the answer should looks like the one for $H_*(MU; \mathbb F_p)$. Namely,

\begin{equation} H_*(MSU; \mathbb F_p) \cong \mathbb F_p[\xi_1, \xi_2, \dots]\otimes_{\mathbb F_p}PH_*(MSU; \mathbb F_p), \end{equation} where $PH_*(MSU; \mathbb F_p)$ is a subalgebra of primitive elements and isomorphic to $\mathbb F_p[y_i| i \neq p^t - 1, i \geqslant 2]$.

The proof of a similar theorem for $MU$ uses the fact that coaction of any cohomology operation in $H_*(MU; \mathbb F_p)$ is defined by its coaction in cohomology of $\mathbb CP^{\infty}$. And this cannot be generalized for the case of $MSU$.

  • 3
    $\begingroup$ You still have maps $(CP^\infty)^{\wedge n}\to MSU$ given as the Thomification of the stable bundle $L_1\oplus\cdots\oplus L_n + L_1^\vee\otimes\cdots\otimes L_n^\vee - (n+1)$ (i.e. the classifying map of the maximal torus $U(1)^n\to SU(n+1)$). Their pullbacks are jointly injective on cohomology. $\endgroup$ Nov 14 '19 at 8:20
  • $\begingroup$ @Bertram yes, of course, but how does it help? $\endgroup$ Nov 16 '19 at 0:35

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