Some observations on the index of $C_p$-spaces

TL;DR

This paper investigates the numerical indices of spaces with free $C_p$-actions, revealing examples where index and co-index differ, and constructing spaces with large index for odd primes.

Contribution

It provides new examples of non-tidy spaces with free $C_p$-actions, including Stiefel manifolds for p=2 and spaces with arbitrarily large index for odd primes.

Findings

Stiefel manifolds are non-tidy for p=2.

Existence of spaces with co-index 3 and arbitrarily large index for odd primes.

Discrepancy between index and co-index in certain $C_p$-spaces.

Abstract

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.