Topological Complexity of $S^3/Q_8$ as fibrewise L-S category

TL;DR

This paper presents an algorithm and a Python implementation to compute the topological complexity of spherical space forms, specifically demonstrating the method on the space $S^3/Q_8$ involving the quaternion group.

Contribution

It introduces a novel algorithm for calculating the fibrewise L-S category and topological complexity of spherical space forms, with a practical Python code for $S^3/Q_8$.

Findings

Successfully computed the topological complexity of $S^3/Q_8$

Provided a general algorithm for spherical space forms

Demonstrated the algorithm with Python implementation

Abstract

In 2010, M. Sakai and the first author showed that the topological complexity of a space $X$ coincides with the fibrewise unpointed L-S category of a pointed fibrewise space $\operatorname{pr}_{1} : X \times X \to X$ with the diagonal map $\Delta : X \to X \times X$ as its section. In this paper, we describe our algorithm how to determine the fibrewise L-S category or the Topological Complexity of a topological spherical space form. Especially, for $S^3/Q_8$ where $Q_8$ is the quaternion group, we write a python code to realise the algorithm to determine its Topological Complexity.