  In this paper, we investigate free actions of some compact groups on
cohomology real and complex Milnor manifolds. More precisely, we compute the
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 of a real or a complex Milnor manifold. As applications, we deduce some
Borsuk-Ulam type results for equivariant maps between spheres and these spaces.
For the complex case, we obtain a lower bound on the Schwarz genus, which
further establishes the existence of coincidence points for maps to the
Euclidean plane.
