# Computation of mod p homology of $MSU$

I am trying to proof Novikov theorem

$$$$MSU_*\otimes \mathbb Z[\frac 1 2] \cong \mathbb Z[\frac 1 2][y_2, y_4, \ldots],\quad \deg y_i = 2i.$$$$

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,

$$$$H_*(MSU; \mathbb F_p) \cong \mathbb F_p[\xi_1, \xi_2, \dots]\otimes_{\mathbb F_p}PH_*(MSU; \mathbb F_p),$$$$ where $$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 $$MSU$$.

• 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. Nov 14 '19 at 8:20
• @Bertram yes, of course, but how does it help? Nov 16 '19 at 0:35