On the K-theory of SO(n) (Q802167)

Let SO(n) be the rotation group of degree n. The purpose of the present paper is to compute \(K^*_{{\mathbb{R}}}(SO(n))\) for \(n\equiv -1,0,1 mod 8\) and \(K^*_{{\mathbb{C}}}(SO(n))\) as algebras. The calculations are based on the theorem of \textit{L. Hodgkin} in Topology 6, 1-36 (1967; Zbl 0186.571) for the \(K_{{\mathbb{C}}}\)-group of the spinor group Spin(n) and the theorem of \textit{R. M. Seymour} in Q. J. Math., Oxford II. Ser. 24, 7-30 (1973; Zbl 0258.57021) for the \(K_{{\mathbb{R}}}\)-group of Spin(n). The author shows that there exists a short exact sequence in the equivariant \(K_{\Lambda}\)-theory involving the injection \(K^*_{\Lambda}(SO(n))\to K^*_{\Lambda}(P^{n-1})\otimes K^*_{\Lambda}(Spin(n))\) where \(\Lambda ={\mathbb{C}}\), or \(\Lambda ={\mathbb{R}}\) and \(n\equiv -1,0,1 mod 8\) and where \(P^ m\) is the real projective m-space, and using this exact sequence he gives the proofs of his results.