The complex \(K^\ast\) ring of the complex projective Stiefel manifold (Q2187741)

The complex projective Stiefel manifold \(PW_{n,k}\) is the quotient of the complex Stiefel manifold \(W_{n,k}\) (of orthonormal \(k\)-frames in \(\mathbb C^n\)) under the \(S^1\)-action given by \(g\cdot(v_1,\ldots,v_k)=(gv_1,\ldots,gv_k)\). The present paper is devoted to computing the complex \(K^*\) ring of this manifold. The author actually works with a slight generalization of \(PW_{n,k}\) and uses the Hodgkin spectral sequence to obtain the following description of \(K^*(PW_{n,k})\): \[ K^*(PW_{n,k})=\Lambda_{\mathbb Z}^*(w_0,\ldots,w_{k-2})\otimes_{\mathbb Z}\mathbb Z[z]/\langle b\rangle, \] where \(w_0,\ldots,w_{k-2}\) are of degree one, \(z\) is of degree zero, and \(b=\gcd(\binom{n}{i}z^i\mid n-k<i\leq n)\).