KO-theory of complex Grassmannians
Let \(M_{m,n}\) be the complex Grassmann manifold \(G_ m(C^{m+n})\) of \(m\)-planes in \(C^{m+n}\). There is a homeomorphism: \(M_{m,n}\buildrel\simeq\over \longrightarrow U(m+n)/U(m)\times U(n)\). The \(KO^ i\)-groups of \(M_{m,n}\) are studied by \textit{S. G. Hoggar} [Q. J. Math., Oxf. II. Ser. 20, 447-463 (1969; Zbl 0184.484)]. The free parts of them are determined, but the torsion parts are partially known [\textit{Hoggar}, loc. cit.; \textit{S. A. Ilori}, K-Theory 2, No. 5, 623-624 (1989; Zbl 0671.55005)]. Here we compute them for arbitrary \(m\) and \(n\), using only the Atiyah-Hirzebruch spectral sequence.
scientific article; zbMATH DE number 59745