Topological complexity of \(S^{3}/Q_{8}\) as fibrewise L-S category. (English) Zbl 1539.55002
The topological complexity \(\mbox{TC}(X)\) of a space \(X\) refers to the sectional category, or Schwarz genus, of the end-points evaluation fibration \(\pi: \mathcal{P}(X)\rightarrow X\times X,\) where \(\pi (\alpha ) = (\alpha (0), \alpha (1))\). Here, \(\mathcal{P}(X)\) represents the set of continuous maps \(\mbox{Map}([0,1],X)\) equipped with the compact-open topology. This concept, introduced by M. Farber [Discrete Comput. Geom. 29 No. 2, 211–221 (2003; Zbl 1038.68130)], has its origins in Robotics, particularly in the motion planning problem. The computation of this homotopy invariant is notoriously challenging.
In [Topology Appl. 157, No. 1, 10–21 (2010); erratum ibid. 159, No. 10–11, 2810–2813 (2012; Zbl 1192.55002)], N. Iwase and M. Sakai established a significant result. They showed that the topological complexity of a space \(X\) aligns with the fibrewise unpointed L-S category of the pointed fibrewise space \(pr_1: X\times X\rightarrow X\), where the diagonal map \(\Delta :X\rightarrow X\times X\) serves as its section.
In the focus of the paper at hand, the authors leverage this connection between topological complexity and fibrewise LS category. They employ fibrewise techniques to determine the topological complexity of \(S^3/Q_8\), a spherical space form derived as the orbit space of the unit sphere of \(\mathbb{H}^t\), where \(t\) is a non-negative integer and \(\mathbb{H}\) denotes the set of quaternionic numbers. This space form emerges from the diagonal action of the subgroup \(Q_8=\langle a,b\mid a^4=b^4=abab^{-1}=1,b^2=a^2\rangle\) of \(\mbox{Sp}(1)\). The authors achieve the computation of this topological complexity, establishing \(\mbox{TC}(S^3/Q_8)=6\). Notably, the paper includes an algorithm and Python code instrumental in this calculation.
In [Topology Appl. 157, No. 1, 10–21 (2010); erratum ibid. 159, No. 10–11, 2810–2813 (2012; Zbl 1192.55002)], N. Iwase and M. Sakai established a significant result. They showed that the topological complexity of a space \(X\) aligns with the fibrewise unpointed L-S category of the pointed fibrewise space \(pr_1: X\times X\rightarrow X\), where the diagonal map \(\Delta :X\rightarrow X\times X\) serves as its section.
In the focus of the paper at hand, the authors leverage this connection between topological complexity and fibrewise LS category. They employ fibrewise techniques to determine the topological complexity of \(S^3/Q_8\), a spherical space form derived as the orbit space of the unit sphere of \(\mathbb{H}^t\), where \(t\) is a non-negative integer and \(\mathbb{H}\) denotes the set of quaternionic numbers. This space form emerges from the diagonal action of the subgroup \(Q_8=\langle a,b\mid a^4=b^4=abab^{-1}=1,b^2=a^2\rangle\) of \(\mbox{Sp}(1)\). The authors achieve the computation of this topological complexity, establishing \(\mbox{TC}(S^3/Q_8)=6\). Notably, the paper includes an algorithm and Python code instrumental in this calculation.
Reviewer: José Calcines (La Laguna)
MSC:
| 55M30 | Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects) |
| 55R70 | Fibrewise topology |
| 55M35 | Finite groups of transformations in algebraic topology (including Smith theory) |
| 55P35 | Loop spaces |
| 57T30 | Bar and cobar constructions |
Software:
PythonReferences:
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