The spaces $O$ and $O/U$ that appear in Bott periodicity represent the functors $KO^7(X)$ and $KO^6(X)$ respectively. Is there an interpretation of the map $KO^7(X) \to KO^6(X)$ induced by the quotient map $O \to O/U$? Or is it always zero?
What about $KO^3(X) \to KO^2(X)$ induced by $Sp \to Sp/U$?