\(KO\)-rings of \(S^{2k+1}/\mathbb{Z} z_{2^n}\) (Q1382117)

Let \(L_k(2^n)\) be the quotient \(S^{2k+1}/Z_{2^n}\), where the group \(Z_{2^n}\) of complex \(2^n\)th roots of unity acts on unit sphere \(S^{2k+1}\) in \(C^{k+1}\) in the usual way: \(z(z_0, \dots, z_k) =(zz_0, \dots, zz_k)\). In this note the author gives generators and relations for the real \(K\)-group \(KO(L_k (2^n))\), \(n\geq 2\). Using this result the author also determines the orders of powers of the element in \(KO(L_k (2^n))\) which arises from the canonical complex line bundle over \(L_k(2^n)\).