The real K-groups of SO(n) for n\(\equiv 2\) mod 4 (Q916140)

In a previous paper by the author [ibid. 25, 185-211 (1988; Zbl 0648.55008)], he determined the algebraic structure of \(KO^*(SO(n))\) for \(n\equiv 1,3,4 mod 4\). The aim of this note is to study the remaining case, that is for \(n\equiv 2 mod 4\). The map from \(P^{n-1}\times Spin(n)\) to SO(n) induces a morphism in KO-theory \(I: KO^*(SO(n))\to KO^*(P^{n-1}\times Spin(n)).\) Using the results on the algebraic structure of \(KO^*(SO(n))\) for \(n\equiv 1 mod4\), the author determines the kernel and image of I and he computes \(KO^*(SO(n))\) for \(n\equiv 2 mod 4\), from \(KO^*(P^{n-1}\times Spin(n))\).