Some observations on the index of $C_p$-spaces

In this paper, we consider numerical indices associated to spaces with free $C_p$-action. We prove that the Stiefel manifolds provide an example of non-tidy spaces for $p=2$, which are those whose co-index and index disagree. In the case of odd primes, we construct examples of co-index $3$ whose index may be arbitrarily large.