On equivariant and invariant topological complexity of smooth $\mathbb{Z}/p$-spheres

TL;DR

This paper studies the equivariant and invariant topological complexity of smooth spheres with cyclic group actions, providing exact values in many cases and showing these invariants can be arbitrarily high.

Contribution

It determines the invariants for semilinear spheres and provides examples demonstrating their unboundedness in smooth cases.

Findings

Semilinear $Z/p$-spheres have invariants 2 or 3.

Exact values are computed for all but two linear action cases.

Invariants can be arbitrarily high for smooth $Z/p$-spheres.

Abstract

We investigate equivariant and invariant topological complexity of spheres endowed with smooth non-free actions of cyclic groups of prime order. We prove that semilinear $\mathbb{Z}/p$-spheres have both invariants either $2$ or $3$ and calculate exact values in all but two cases for linear actions. On the other hand, we exhibit examples which show that these invariants can be arbitrarily high in the class of smooth $\mathbb{Z}/p$-spheres.