Using pairing in Atiyah-Hirzebruch spectral sequence one can show that homology of $BU(n)$ is a free abelian group with basis $\alpha_{k_1}\cdots\alpha_{k_t}$, $k\leqslant n$, where $\alpha_{i} = \big((c_1)^i\big)^* \in H_{2i}(\mathbb C P^\infty)$. Therefore, passing to (co)limits and applying the Thom isomorphism, we have \begin{equation} H_*(MU) \cong \mathbb Z[a_1, a_2, \ldots]. \end{equation}

Since polynomial generators come as image of homology classes of $\mathbb C P^\infty$, $\mathfrak A_p^*$-comodule structure of $H_*(MU; \mathbb F_p)$ can be deduced from the one of $H_*(\mathbb C P^\infty; \mathbb F_p)$.

Is there any similar description of $H_*(MSU; \mathbb F_p)$?


Let $S = \mathbb F_p[x_n| n+1\neq p^t]$. Define $f\colon H_*(MU;\mathbb F_p) \to S$ by the formula \begin{equation*} f(a_n) = \begin{cases} x_n;\quad&\text{if $n+1 \neq p^t$;} \\ 0;&\text{otherwise.} \end{cases} \end{equation*} Then the composition \begin{equation*} H_*(MU;\mathbb F_p) \xrightarrow{\text{coaction}} \mathfrak A_p'\otimes_{\mathbb F_p} H_*(MU;\mathbb F_p) \xrightarrow{\mathrm{id}\otimes f} \mathfrak A_p'\otimes_{\mathbb F_p} S \end{equation*} is an isomorphism of $\mathbb F_p$-algebras and $\mathfrak A_p^*$-comodules. Here $\mathfrak A_p' = \mathbb F_p[\xi_1, \xi_2, \ldots]$ is the quotient of the dual Steenrod algebra $\mathfrak A_p^*$.

The Adams' paper Primitive Elements in the $K$-theory of $BSU$ contains a computation of homology $H_*(BSU; \mathbb Z)$. Namely, $H_*(BSU; \mathbb Z) \cong \mathbb Z[y_2, y_3, \ldots]$, $\deg y_i = 2i$. And therefore, $H_*(MSU; \mathbb Z) \cong \mathbb Z[Y_2, Y_3, \ldots]$, $\deg Y_i = 2i$. But still the proof does not describe the embedding \begin{equation*} H_*(MSU; \mathbb Z) \hookrightarrow H_*(MU; \mathbb Z). \end{equation*}

Are there any explicit formulae for the images of $y_n$ in $H_*(MU)$? I would be really grateful for something explicit that could be useful for describing the $\mathfrak A_p^*$-comodule structure of $H_*(MSU;\mathbb F_p)$.

  • 2
    $\begingroup$ Basically you can play the same game at odd prime. I don't know about the explicit $F_p$ algebra structure, but $A_p$ commodule structure at odd prime is quite simple, and at 2, it is described in (1.5) of The Homotopy Type of MSU David J. Pengelley American Journal of Mathematics American Journal of Mathematics Vol. 104, No. 5 (Oct., 1982), pp. 1101-1123 $\endgroup$
    – user43326
    Mar 12, 2020 at 8:02
  • $\begingroup$ @user43326, is there any reference to $A_p^*$-comodule structure of $H_*(MSU;\mathbb F_p)$ for odd $p$? $\endgroup$ Mar 12, 2020 at 11:47
  • $\begingroup$ I don't have any reference, but at odd $p$, $MSU$ splits as wedge of $BP$, this gives the action of the Steenrod algebra. $\endgroup$
    – user43326
    Mar 12, 2020 at 14:27
  • $\begingroup$ @user43326 do you how to prove the latter? $\endgroup$ Mar 12, 2020 at 15:22
  • $\begingroup$ Well, Pengalley says, "standard method"... I would imagine you can simply compute it using the Thom isomorphism. Although there might be an argument using the fact that both $MSU$'s homotopy and homology are torsion-free. $\endgroup$
    – user43326
    Mar 13, 2020 at 7:10


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