Topological Complexity of $S^3\mathrm{/}{Q}_8$ as fibrewise L-S category

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 ${\mathrm{pr}}_1:X\times X\to X$ with the diagonal map $\mathrm{\Delta }:X\to X\times X$ 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 $S^3\mathrm{/}{Q}_8$ where $Q_8$ is the quaternion group, we write a python code to realise the algorithm to determine its Topological Complexity.