Topological complexity of $S^3/Q_8$ as fibrewise L-S category
DOI:
https://doi.org/10.12775/TMNA.2022.068Keywords
Topological complexity, space form, quaternion group, python programAbstract
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.References
A. Adem and R.J. Milgram, Cohomology of finite groups, Second, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 309, Springer–Verlag, Berlin, 2004.
D.J. Benson, Representations and Cohomology, II: Cohomology of Groups and Modules, Cambridge Studies in Advanced Mathematics, vol. 31, Cambridge University Press, Cambridge, 1991.
M. Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), no. 2, 211–221.
M. Farber and M. Grant, Robot motion planning, weights of cohomology classes, and cohomology operations, Proc. Amer. Math. Soc. 136 (2008), no. 9, 3339–3349.
K. Fujii, On the K-ring of S 4n+3 /Hm , Hiroshima Math. J. 3 (1973), 251–265.
N. Iwase and M. Sakai, Topological complexity is a fibrewise L- category, Topology Appl. 157 (2010), no. 1, 10–21.
N. Iwase, M. Sakai and M. Tsutaya, A short proof for tc(K) = 4, Topology Appl. 264 (2019), 167–174.
I.M. James, Introduction to fibrewise homotopy theory, Handbook of Algebraic Topology, 1995, pp. 169–194.
G. Segal, Categories and cohomology theories, Topology 13 (1974), 293–312.
K. Shimakawa, K. Yoshida and T. Haraguchi, Homology and cohomology via enriched bifunctors, Kyushu J. Math. 72 (2018), no. 2, 239–252.
G.W. Whitehead, Elements of Homotopy Theory, Graduate Texts in Mathematics, vol. 61, Springer–Verlag, New York, Berlin, 1978.
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Norio Iwase, Yuya Miyata
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0
