On the topological complexity of $S^3/Q_8$

TL;DR

This paper calculates the topological complexity of the space $S^3/Q_8$ using a fibrewise approach, contributing to the understanding of this invariant for specific quotient spaces.

Contribution

It introduces a method based on fibrewise topology to determine the topological complexity of $S^3/Q_8$, expanding concrete examples of this invariant.

Findings

Topological complexity of $S^3/Q_8$ determined explicitly

Fibrewise approach effectively applied to quotient spaces

Enhances understanding of TC in specific topological contexts

Abstract

Topological complexity was first introduced in 2003 by Michael Farber as a homotopy invariant for a connected topological space X, denoted by TC(X). Although the invariant is defined in terms of elementary homotopy theory using well-known Serre path fibration, not many examples are known to be determined concretely by now. In 2010, Iwase and Sakai showed that the topological complexity of a space is a fibrewise version of a L-S category for a fibrewise space over the space. In this paper, we determine the topological complexity of $S^3/Q_8$ using a method produced from the fibrewise view point.