  For finite dimensional free $C_p$-spaces, the calculation of the Bredon
cohomology ring as an algebra over the cohomology of $S^0$ is used to prove the
non-existence of certain $C_p$-maps. These are related to Borsuk-Ulam type
theorems, and equivariant maps related to the topological Tverberg conjecture.
For certain finite dimensional $C_p$-spaces which are formed out of
representations, it is proved that the cohomology is a free module over the
cohomology of a point. All the calculations are done for the cohomology with
constant coefficients $\mathbb{Z}/p$.
