All cohomology and homology will be $Z/2$ coefficient. The restriction map $H^*(BSO(3))\rightarrow H^*(BO(2))$ is well-known to be the inclusion of the Dickson invariant $Z/2[w_2,w_3]$ into the symmetric polynomial algebra $Z/2[w_1,w_2]$. It is also well-known that the transfer $H^*(BO(2)) \rightarrow H^*(BSO(3))$ gives a section.
What is less known is the evaluation of the transfer, or the composition $H^*(BO(2))\rightarrow H^*(BSO(3))\rightarrow H^*(BO(2))$, or the dual $H_*(BO(2))\rightarrow H_*(BSO(3))\rightarrow H_*(BO(2))$. Of course, this follows easily from Feshbach "The transfer and compact Lie groups" Transactions of the American Mathematical Society vol 251 (1979) pp.139-169. Theorem II.11, and in cohomology one gets $$f(x_1,x_2)\rightarrow f(x_1,x_2)+f_(x_1+x_2,x_2)+f(x_1+x_2,x_1)$$
By ``dualizing'' in homology one gets $$A(s)\cdot A(t)\rightarrow A(s)\cdot A(t) + A(s+t)\cdot (A(s)+A(t))$$ where we set $A(s)=\Sigma a_is^i$ with $a_i$ generator of $H_i(BO(1))$, $A(X)=\Sigma _ia_iX^i$, and $\cdot$ denote s the "concatenation product'' $H_*(BO(1)^2)\rightarrow H_*(BO(2))$.
As I wrote, this is not difficult, but as we see so often $H^*(BSO(3))$ and $H^*(BO(2))$ in the litterature, I feel that this should appear somewhere, but so far I haven't been able to find the reference. Does anyone know if there is any published reference of this? Thank you very much in advance.
