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

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

• I am not sure that it has exactly what you are looking for, but Neil Strickland's thesis contains an astounding amount of information about the homology of K-theoretic spaces, such as those in your question. Even if it does not answer your question, it might help you with many related questions. – Justin Noel Oct 25 '13 at 8:02

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.

• Thanks @Oscar! I'll try to work it out from here. I'll accept your answer once I do, and post a full solution for posterity! – Jonathan Beardsley Oct 25 '13 at 17:55
• So I think the relevant reference is <a href="bit.ly/17j9KfW"> this </a>. – Jonathan Beardsley Oct 25 '13 at 20:02
• Although I'm not sure that the generators on the homology of BO that Pengelley gives are the same as your generators. – Jonathan Beardsley Oct 25 '13 at 20:56
• So I feel bad about this, but in fact, the structure of the homology of MSO inside of the homology of MO, in terms of the Stiefel-Whitney classes, does not seem to be known (and appears to be rather hard to work out). So I'm not sure how to extend this to a full answer. – Jonathan Beardsley Oct 29 '13 at 5:04
• But if you don't have a description of the homology of BSO, what possible form can an answer to your question have? – Oscar Randal-Williams Oct 29 '13 at 7:59