Integral (co)homology of $SU/SO$

Question

I would like to know the integral cohomology of $SU(\infty)/SO(\infty)$ (to degree 5 or 6, say.)

Mimura-Toda says $H^*(SU/SO,\mathbb{Z}/2\mathbb{Z})=\wedge[w_2,w_3,\ldots]$ where $w_i$ is a pullback of Stiefel-Whitney classes via $SU/SO\to BSO$.

I'd like to know the image of Hurewicz images, too. Bott periodicity basically says $\pi_2(SU/SO)=\pi_3(SU/SO)=\mathbb{Z}/2\mathbb{Z}$ and I would like to know their images in $H_*(SU/SO,\mathbb{Z})$.

I feel stupid for asking this, because I can find many results on exceptional symmetric spaces by googling, while I do not find it for the non-exceptional symmetric spaces...

Answer 1

First, I'll mostly talk about $U/O$ rather than $SU/SO$ because $U/O$ can be descibed as $B(\mathbb{Z}\times BO)$ or as the $8k-1$'th space in the $\Omega$-spectrum for $KO$. This gives $\pi_0(U/O)=0$ and $\pi_1(U/O)=\pi_0(KO)=\mathbb{Z}$. From the Hurewicz and universal coefficient theorems this gives $H^1(U/O)=\text{Hom}(\pi_1(U/O,\mathbb{Z})$. This describes $[S^1,U/O]=\pi_1(U/O)$ and $[U/O,S^1]=[U/O,K(\mathbb{Z},1)]=H^1(U/O)$ and using this together with the $H$-space structure we obtain a splitting of $U/O$ as the product of $S^1$ with the universal cover, which is $SU/SO$. So there is not too much difference between $U/O$ and $SU/SO$.

The ring $H_*(U/O;\mathbb{Z}/2)$ has generators $a_k$ of degree $k$ for $k>0$, satisfying $a_{2k}=a_k^2$. We can therefore discard the even generators and say that $H_*(U/O;\mathbb{Z}/2)$ is polynomial on the classes $a_{2k+1}$. The Bockstein operation is given by $\beta(a_{2k+1})=a_{2k}$ and $\beta(a_{2k+2})=0$. (This would be more awkward to state if we had not introduced the classes $a_k$ for all $k$.) It follows that the classes $h_{4k+1}=a_{4k+1}+a_{2k}a_{2k+1}$ satisfy $\beta(h_{4k+1})=0$. In fact one can show that the ring $H_*(H_*(U/O;\mathbb{Z}/2),\beta)=\ker(\beta)/\text{img}(\beta)$ is an exterior generated by these classes $h_{4k+1}$ (including $h_1$, which should be interpreted as $a_1$). One can also show that there are classes $b_{4k+1}\in H_{4k+1}(U/O)$ lifting $h_{4k+1}$, and that $H_*(U/O)/\text{torsion}$ is an exterior algebra generated by these elements. The torsion is all annihilated by $2$ and is the image of the Bockstein map $\beta'\colon H_i(U/O;\mathbb{Z}/2)\to H_{i-1}(U/O)$. It is awkward to give a full description of the torsion, but if you are only interested in low degrees you should be able to spell it out.

I am reading all this off from my thesis, where everything is done using Hopf ring methods. Certainly some parts of the calculation appear already here:

Henri Cartan, Périodicité des groupes d’homotopie stables des groupes classiques, d’après Bott, Sem. H. Cartan, vol. 60, Ecole Normale Supérieur, 1959.

However, I don't have that to hand, so I don't remember precisely what is covered.