Abstract
We calculate the higher topological complexity TC s for the complements of reflection arrangements, in other words for the pure Artin type groups of all finite complex reflection groups. In order to do that we introduce a simple combinatorial criterion for arrangements sufficient for the cohomological lower bound for TC s to coincide with the dimensional upper bound.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
V.I. Arnol’d, The cohomology ring of the group of dyed braids. Math. Notes 5, 138–140 (1969)
I. Basabe, J. Gonzalez, Y. Rudyak, D. Tamaki, Higher topological complexity and its symmetrization. Algebr. Geom. Topology 14, 2103–2124 (2014)
D. Bessis, Finite complex reflection arrangements are K(π, 1) (2006). arXiv:math.GT/0610777
D. Cohen, G. Pruidze, Motion planning in tori. Bull. Lond. Math. Soc. 40, 249–262 (2008)
P. Deligne, Les immeubles des groupes de tresses generalises. Invent. Math. 17, 273–302 (1972)
M. Farber, Topological complexity of motion planning. Discrete Comput. Geom. 29, 211–221 (2003)
M. Farber, S. Yuzvinsky, Topological robotics: subspace arrangements and collision free motion planning. Geometry, Topology, and Mathematical Physics. American Mathematical Society Translations Series 2, vol. 212 (American Mathematical Society, Providence, 2000), pp. 145–156
J. Gonzalez, M. Grant, Sequential motion planning of non-colliding particles in Euclidean spaces (2013). arXiv:math1309.4346
P. Orlik, L. Solomon, Unitary reflection groups and cohomology. Invent. Math. 59, 77–94 (1980)
P. Orlik, H. Terao, Arrangements of Hyperplanes. Grundlehren der Mathematischen Wissenschaften, vol. 300 (Springer, Berlin/Heidelberg/New York, 1992)
Y. Rudyak, On higher analogs of topological complexity. Topology Appl. 157, 916–920 (2010)
A. Shwarz, The genus of a fiber space. AMS Transl. 23 (14), 5339–5354 (1995)
S. Yuzvinsky, Topological complexity of generic hyperplane complements. Contemp. Math. 438, 115–119 (2007)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Yuzvinsky, S. (2016). Higher Topological Complexity of Artin Type Groups. In: Callegaro, F., Cohen, F., De Concini, C., Feichtner, E., Gaiffi, G., Salvetti, M. (eds) Configuration Spaces. Springer INdAM Series, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-319-31580-5_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-31580-5_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-31579-9
Online ISBN: 978-3-319-31580-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)