The Image of the Mod 2 Homology of BSp in the Homology of BSO

Question

I'm essentially trying to figure out exactly what the title asks for. I've been scouring old Seminaires Henri Cartan and books by Stong to try to see exactly how to do this, but the combination of French and older terminology has made it rough going.

In particular, I've been able to work out (I think!) that, modulo 2, the map $$H_\ast(BSp;\mathbb{Z}/2)\to H_\ast(BU;\mathbb{Z}/2) $$ takes $y_{4k}$ to $(x_{2k})^2$, and that the map

$$H_\ast(U;\mathbb{Z}/2)\to H_\ast(SO;\mathbb{Z}/2)$$

takes $a_{2k-1}$ to certain polynomials $t_{2k-1}$ (the primitives) defined recursively as $$t_1=c_1,t_3=c_3+c_1c_2,\ldots,t_{2k-1}=\sum_{\substack{i+j=2k+1\\0\leq i<j}}c_ic_j$$ where the $a_{2k-1}$ and $c_k$ generate the homology groups. I'd really appreciate anything, from a complete exposition, to a reference, to a hint about something that I've missed or gotten wrong. I'm obviously especially interested in working out how to lift that last map to the classifying spaces (which would require completely knowing what the so-called "suspension" maps look like).

Thanks so much!

-Jon

Answer 1

We have $H^*(BO, \mathbb{F}_2) = \mathbb{F}_2[w_1, w_2, \ldots]$, where the $w_i$ are the Stiefel--Whitney classes. If $f: \mathbb{RP}^\infty \to BO$ classifies the reduced universal real line bundle, and $e_i \in H^i(\mathbb{RP}^\infty;\mathbb{F}_2)$ is the nontrivial class, we let $a_i = f_*(e_i)$ and the Pontrjagin ring structure on $H_*(BO;\mathbb{F}_2)$ is then $\mathbb{F}_2[a_1, a_2, \ldots]$.

Similarly $H^*(BSp, \mathbb{F}_2) = \mathbb{F}_2[k_1, k_2, \ldots]$, where the $k_i$ are the symplectic Pontrjagin classes (of degree $4i$). If $g: \mathbb{HP}^\infty \to BSp$ classifies the reduced universal quaternionic line bundle, and $d_i \in H^{4i}(\mathbb{HP}^\infty;\mathbb{F}_2)$ is the nontrivial class, we let $b_i = g_*(d_i)$ and the Pontrjagin ring structure on $H_*(BSp;\mathbb{F}_2)$ is then $\mathbb{F}_2[b_1, b_2, \ldots]$.

If $\phi: BSp \to BO$ is the realification map, everything you need to know is packaged into the formula $$\phi^*(w_i) = \begin{cases} k_{i/4} & i \equiv 0 (4)\\ 0 & \text{else}, \end{cases}$$ which is better expressed as $\phi^*w = k$: the pullback of the total Stiefel--Whitney class is the total symplectic Pontrjagin class.

You want to compute $\phi_*(b_i) = (\phi \circ g)_*(d_i) \in \mathbb{F}_2[a_1, a_2, \ldots]$. From the above it is easy to see that $$\langle \phi_*(b_i), w_I\rangle = \begin{cases} 1 & w_I = w_4^i\\ 0 & \text{else}, \end{cases}$$ and so $\phi_*(b_i) = a_i^4$.

Thus the image in $H_*(BO;\mathbb{F}_2) = \mathbb{F}_2[a_1, a_2, \ldots]$ is $\mathbb{F}_2[a_1^4, a_2^4, \ldots]$. I don't know how the homology of $BSO$ is best described inside that of $BO$, but presumably one can find out.