Free actions of some compact groups on Milnor manifolds

In this paper, we investigate free actions of some compact groups on mod 2 cohomology real and complex Milnor manifolds. More precisely, we compute the possible mod 2 cohomology algebra of the orbit space of an arbitrary free $mathbb{Z}_2$ and $mathbb{S}^1$-action on a compact Hausdorff space with mod 2 cohomology algebra isomorphic to that of a real or a complex Milnor manifold. As applications, we deduce some Borsuk-Ulam type results.