## Algebraic & Geometric Topology

### Bredon cohomology and robot motion planning

#### Abstract

We study the topological invariant $TC(X)$ reflecting the complexity of algorithms for autonomous robot motion. Here, $X$ stands for the configuration space of a system and $TC(X)$ is, roughly, the minimal number of continuous rules which are needed to construct a motion planning algorithm in $X$. We focus on the case when the space $X$ is aspherical; then the number $TC(X)$ depends only on the fundamental group $π=π1(X)$ and we denote it by $TC(π)$. We prove that $TC(π)$ can be characterised as the smallest integer $k$ such that the canonical $π×π$–equivariant map of classifying spaces

$E ( π × π ) → E D ( π × π )$

can be equivariantly deformed into the $k$–dimensional skeleton of $ED(π×π)$. The symbol $E(π×π)$ denotes the classifying space for free actions and $ED(π×π)$ denotes the classifying space for actions with isotropy in the family $D$ of subgroups of $π×π$ which are conjugate to the diagonal subgroup. Using this result we show how one can estimate $TC(π)$ in terms of the equivariant Bredon cohomology theory. We prove that $TC(π)≤ max{3,cdD(π×π)}$, where $cdD(π×π)$ denotes the cohomological dimension of $π×π$ with respect to the family of subgroups $D$. We also introduce a Bredon cohomology refinement of the canonical class and prove its universality. Finally we show that for a large class of principal groups (which includes all torsion-free hyperbolic groups as well as all torsion-free nilpotent groups) the essential cohomology classes in the sense of Farber and Mescher (2017) are exactly the classes having Bredon cohomology extensions with respect to the family $D$.

#### Article information

Source
Algebr. Geom. Topol., Volume 19, Number 4 (2019), 2023-2059.

Dates
Revised: 29 December 2018
Accepted: 15 January 2019
First available in Project Euclid: 22 August 2019

https://projecteuclid.org/euclid.agt/1566439280

Digital Object Identifier
doi:10.2140/agt.2019.19.2023

Mathematical Reviews number (MathSciNet)
MR3995023

Subjects
Secondary: 55M99: None of the above, but in this section

#### Citation

Farber, Michael; Grant, Mark; Lupton, Gregory; Oprea, John. Bredon cohomology and robot motion planning. Algebr. Geom. Topol. 19 (2019), no. 4, 2023--2059. doi:10.2140/agt.2019.19.2023. https://projecteuclid.org/euclid.agt/1566439280

#### References

• A Boudjaj, Y Rami, On spaces of topological complexity two, preprint (2016)
• G E Bredon, Equivariant cohomology theories, Lecture Notes in Mathematics 34, Springer (1967)
• K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer (1982)
• D C Cohen, G Pruidze, Motion planning in tori, Bull. Lond. Math. Soc. 40 (2008) 249–262
• D C Cohen, L Vandembroucq, Topological complexity of the Klein bottle, J. Appl. Comput. Topol. 1 (2017) 199–213
• O Cornea, G Lupton, J Oprea, D Tanré, Lusternik–Schnirelmann category, Mathematical Surveys and Monographs 103, Amer. Math. Soc., Providence, RI (2003)
• A Costa, M Farber, Motion planning in spaces with small fundamental groups, Commun. Contemp. Math. 12 (2010) 107–119
• T tom Dieck, Transformation groups, De Gruyter Studies in Mathematics 8, de Gruyter, Berlin (1987)
• A Dranishnikov, The topological complexity and the homotopy cofiber of the diagonal map for non-orientable surfaces, Proc. Amer. Math. Soc. 144 (2016) 4999–5014
• A N Dranishnikov, Y B Rudyak, On the Berstein–Svarc theorem in dimension $2$, Math. Proc. Cambridge Philos. Soc. 146 (2009) 407–413
• S Eilenberg, T Ganea, On the Lusternik–Schnirelmann category of abstract groups, Ann. of Math. 65 (1957) 517–518
• M Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003) 211–221
• M Farber, Topology of robot motion planning, from “Morse theoretic methods in nonlinear analysis and in symplectic topology” (P Biran, O Cornea, F Lalonde, editors), NATO Sci. Ser. II Math. Phys. Chem. 217, Springer (2006) 185–230
• M Farber, Invitation to topological robotics, Eur. Math. Soc., Zürich (2008)
• M Farber, Configuration spaces and robot motion planning algorithms, from “Combinatorial and toric homotopy” (A Darby, J Grbic, Z Lü, J Wu, editors), Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 35, World Sci., Hackensack, NJ (2018) 263–303
• M Farber, M Grant, G Lupton, J Oprea, An upper bound for topological complexity, Topology Appl. 255 (2019) 109–125
• M Farber, S Mescher, On the topological complexity of aspherical spaces, preprint (2017)
• M Farber, J Oprea, Higher topological complexity of aspherical spaces, Topology Appl. 258 (2019) 142–160
• M Farber, S Tabachnikov, S Yuzvinsky, Topological robotics: motion planning in projective spaces, Int. Math. Res. Not. 2003 (2003) 1853–1870
• M Farber, S Yuzvinsky, Topological robotics: subspace arrangements and collision free motion planning, from “Geometry, topology, and mathematical physics” (V M Buchstaber, I M Krichever, editors), Amer. Math. Soc. Transl. Ser. 2 212, Amer. Math. Soc., Providence, RI (2004) 145–156
• M Fluch, On Bredon (co-)homological dimensions of groups, PhD thesis, University of Southhampton (2011)
• M Grant, Topological complexity, fibrations and symmetry, Topology Appl. 159 (2012) 88–97
• M Grant, G Lupton, J Oprea, Spaces of topological complexity one, Homology Homotopy Appl. 15 (2013) 73–81
• M Grant, G Lupton, J Oprea, A mapping theorem for topological complexity, Algebr. Geom. Topol. 15 (2015) 1643–1666
• M Grant, G Lupton, J Oprea, New lower bounds for the topological complexity of aspherical spaces, Topology Appl. 189 (2015) 78–91
• M Grant, D Recio-Mitter, Topological complexity of subgroups of Artin's braid groups, from “Topological complexity and related topics” (M Grant, G Lupton, L Vandembroucq, editors), Contemp. Math. 702, Amer. Math. Soc., Providence, RI (2018) 165–176
• D Husemoller, Fibre bundles, McGraw-Hill, New York (1966)
• W Lück, Transformation groups and algebraic $K$–theory, Lecture Notes in Mathematics 1408, Springer (1989)
• W Lück, Survey on classifying spaces for families of subgroups, from “Infinite groups: geometric, combinatorial and dynamical aspects” (L Bartholdi, T Ceccherini-Silberstein, T Smirnova-Nagnibeda, A Zuk, editors), Progr. Math. 248, Birkhäuser, Basel (2005) 269–322
• S Mac Lane, Homology, Grundl. Math. Wissen. 114, Academic, New York (1963)
• J P May, Equivariant homotopy and cohomology theory, CBMS Regional Conference Series in Mathematics 91, Amer. Math. Soc., Providence, RI (1996)
• J Milnor, On spaces having the homotopy type of a ${\rm CW}$–complex, Trans. Amer. Math. Soc. 90 (1959) 272–280
• G Mislin, Equivariant $K$–homology of the classifying space for proper actions, from “Proper group actions and the Baum–Connes conjecture”, Birkhäuser, Basel (2003) 1–78
• Y Rudyak, On topological complexity of Eilenberg–MacLane spaces, Topology Proc. 48 (2016) 65–67
• A S Schwarz, The genus of a fiber space, Trudy Moskov. Mat. Obšč. 10 (1961) 217–272 In Russian; translated with [Sv662?] in “Eleven papers on topology and algebra”, Amer. Math. Soc. Transl. Ser. 2 55, Amer. Math. Soc., Providence, RI (1966) 49–140
• A S Schwarz, The genus of a fiber space, Trudy Moskov Mat. Obšč. 11 (1962) 99–126 In Russian; translated with [Sv661?] in “Eleven papers on topology and algebra”, Amer. Math. Soc. Transl. Ser. 2 55, Amer. Math. Soc., Providence, RI (1966) 49–140
• G W Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics 61, Springer (1978)