Use the Leray–Hirsch theorem as in Corollary 4D.3 to compute \(H^*(V_n(\mathbb{C}^k); \mathbb{Z}) \approx \Lambda_\mathbb{Z}[x_{2k-2n+1}, x_{2k-2n+3}, \ldots, x_{2k-1}]\) and similarly in the quaternionic case.
Solution
1. \(V_n(\mathbb{C}^k)\) is the Stiefel manifold of \(n\)-frames in \(\mathbb{C}^k\), and it fibers over \(V_{n-1}(\mathbb{C}^k)\) with fiber \(S^{2k-2n+1}\).
the standard fibration \(S^{2k-2n+1} \to V_n(\mathbb{C}^k) \to V_{n-1}(\mathbb{C}^k)\).
2. The Leray–Hirsch theorem applies: the cohomology of the fiber \(S^{2k-2n+1}\) is generated by a class \(x_{2k-2n+1}\) that extends to a class on the total space (the Euler class of the associated bundle).
the fibration satisfies the Leray–Hirsch condition (the fiber cohomology is generated by classes that are restrictions of global classes).
3. Hence \(H^*(V_n(\mathbb{C}^k)) \cong H^*(V_{n-1}(\mathbb{C}^k)) \otimes H^*(S^{2k-2n+1})\).
Leray–Hirsch theorem.
4. Iterating (since \(V_1(\mathbb{C}^k) = S^{2k-1}\)), we get\[H^*(V_n(\mathbb{C}^k)) \cong H^*(S^{2k-1}) \otimes H^*(S^{2k-3}) \otimes \cdots \otimes H^*(S^{2k-2n+1}).\]
3, applied inductively.
5. Each \(H^*(S^{2m-1}) = \Lambda_\mathbb{Z}[x_{2m-1}]\) (exterior algebra on one generator of odd degree \(2m-1\)).
cohomology of an odd sphere.
6. Hence \(H^*(V_n(\mathbb{C}^k)) \cong \Lambda_\mathbb{Z}[x_{2k-2n+1}, x_{2k-2n+3}, \ldots, x_{2k-1}]\).
4 and 5.
7. In the quaternionic case, the fiber is \(S^{4k-4n+3}\) (a quaternionic sphere), so\[H^*(V_n(\mathbb{H}^k)) \cong \Lambda_\mathbb{Z}[x_{4k-4n+3}, x_{4k-4n+7}, \ldots, x_{4k-1}].\]
the same argument with the quaternionic Stiefel manifold, whose fibers are spheres of dimension \(4k - 4n + 3, 4k - 4n + 7, \ldots, 4k - 1\).
8. Q.E.D.
6 and 7.