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

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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 $＼proj_{1} ＼colon X ＼times X ＼to X$ with the diagonal map $＼Delta ＼colon X ＼to X ＼times X$ 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.

[Alternative abstract Notation]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 pr_1 ∶ 𝑋 × 𝑋 → 𝑋 with the diagonal map Δ ∶ 𝑋 → 𝑋 × 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∕_𝑄8 where 𝑄_8 is the quaternion group, we write a python code to realise the algorithm to determine its Topological Complexity.

2020 Mathematics Subject Classification. Primary: 55M30; Secondary: 55R70, 55M35, 55P35, 57T30.

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