Singular cohomology rings of some orbit spaces defined by free involution on \(CP(2m+1)\) (Q615816)

On the complex projective space \(\mathbb CP(2k+1)\), the free involution \(J_k\) sending the point \([z_0,z_1,\dots,z_{2k},z_{2k+1}]\) to \([-\bar z_1, \bar z_0,\dots,-\bar z_{2k+1}, \bar z_{2k}]\) is considered. Given positive integers \(k_1,\dots,k_n\), on the product of the projective spaces \(\mathbb CP(2k_\nu+1)\) we have the diagonal action \(J_{k_1} \times \dots\times J_{k_n}\). The author computes the integral singular cohomology ring of the corresponding quotient space in terms of characteristic classes, the transfer homomorphism and Frobenius reciprocity.