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Higher topological complexity and its symmetrization. (English) Zbl 1348.55005

The ‘topological complexity’ \(\mathrm{TC}(X)\) of a space \(X\) defined by Farber is essentially the Švarc genus of the fibration equivalent to the diagonal \(\Delta: X \to X\times X\). Rudyak extended this to the iterated diagonal \(\Delta_n : X \to X^n\), leading to the notion of ‘higher topological complexity’ \(\mathrm{TC}_n(X)\).
In this paper, the authors compute \(\mathrm{TC}_n(X)\) for important examples (§3). Next they introduce two versions of ‘symmetric topological complexity’: \(\mathrm{TC}_n^S(X)\) and \(\mathrm{TC}_n^\Sigma(X)\), the first one extending a notion of Farber and Grant, the second one being a homotopy invariant (§4). Finally they discuss some considerations about the case of spheres (§5).
The authors prove that \(\mathrm{TC}_n(S^{k_1} \times S^{k_2} \times \dots \times S^{k_m}) = m(n-1)+l\) where \(l\) is the number of even dimensional spheres, extending the well-known fact that \(\mathrm{TC}(S^k) =\) 1 if \(k\) is odd and 2 if \(k\) is even. They also prove that for every closed simply connected symplectic manifold \(M^{2m}\) we have \(\mathrm{TC}_n(M) = nm\).
They define \(\mathrm{TC}_n^\Sigma(X)\) as the least integer \(k\) which satisfies \(X^n = A_0 \cup \dots \cup A_k\), where each \(A_i\) is open, symmmetric and admits a continuous equivariant section \(s_i: A_i \to X^{J_n}\) for the map \(e_n: X^{J_n} \to X^n\) which is the fibration equivalent to the iterated diagonal \(\Delta_n : X \to X^n\). The fibration \(e_n\) induces a fibration at the level of orbit spaces of the action of the symmetric group \(\Sigma_n\) on the configuration spaces: \(\epsilon_n: Y_n(X) = e_n^{-1} (\mathrm{Conf}_n (X))/\Sigma_n \to \mathrm{Braid}_n(X) = \mathrm{Conf}_n(X) / \Sigma_n\). The authors prove that if \(X\) is a Euclidean neighborhood retract, then \(\mathrm{genus}(\epsilon_n) \leq \mathrm{TC}_n^\Sigma(X) \leq \mathrm{genus}(\epsilon_n) + \dots + \mathrm{genus}(\epsilon_2) + n-1\). This suggests the definition (for \(n \geq 2\)): \(\mathrm{TC}_n^S(X) = \mathrm{genus}(\epsilon_n) + \dots + \mathrm{genus}(\epsilon_2) + n-1\).
It is shown that if \(k\) is a positive odd integer, for \(X = S^k\) and \(i \geq 2\), \(\mathrm{genus}(\epsilon_i) \geq 1\); further, \(\mathrm{genus}(\epsilon_i) = 1\) provided \(k = 1\). Finally the authors show that \(\mathrm{TC}_n^S(S^1) = 2(n-1)\), extending the known equality \(\mathrm{TC}_2^S(S^k) = 2\) (valid for any \(k > 0\)).

MSC:

55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
55R80 Discriminantal varieties and configuration spaces in algebraic topology
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References:

[1] I Basabe, J González, Y Rudyak, D Tamaki, Higher topological complexity and its symmetrization, · Zbl 1348.55005
[2] G E Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics 46, Academic Press (1972) · Zbl 0246.57017
[3] O Cornea, G Lupton, J Oprea, D Tanré, Lusternik-Schnirelmann category, Math. Surveys Monographs 103, Amer. Math. Soc. (2003) · Zbl 1032.55001
[4] A Dold, Lectures on algebraic topology, Classics in Mathematics, Springer, Berlin (1995) · Zbl 0872.55001
[5] M Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003) 211 · Zbl 1038.68130 · doi:10.1007/s00454-002-0760-9
[6] M Farber, Instabilities of robot motion, Topology Appl. 140 (2004) 245 · Zbl 1106.68107 · doi:10.1016/j.topol.2003.07.011
[7] M Farber, Topology of robot motion planning (editors P Biran, O Cornea, F Lalonde), NATO Sci. Ser. II Math. Phys. Chem. 217, Springer, Dordrecht (2006) 185 · Zbl 1089.68131 · doi:10.1007/1-4020-4266-3_05
[8] M Farber, Invitation to topological robotics, Zurich Lectures Adv. Math. 10, Eur. Math. Soc. (2008) · Zbl 1148.55011 · doi:10.4171/054
[9] M Farber, M Grant, Symmetric motion planning (editors M Farber, R Ghrist, M Burger, D Koditschek), Contemp. Math. 438, Amer. Math. Soc. (2007) 85 · Zbl 1143.70013 · doi:10.1090/conm/438/08447
[10] M Farber, M Grant, Robot motion planning, weights of cohomology classes, and cohomology operations, Proc. Amer. Math. Soc. 136 (2008) 3339 · Zbl 1151.55004 · doi:10.1090/S0002-9939-08-09529-4
[11] M Farber, S Tabachnikov, S Yuzvinsky, Topological robotics: Motion planning in projective spaces, Int. Math. Res. Not. 2003 (2003) 1853 · Zbl 1030.68089 · doi:10.1155/S1073792803210035
[12] M Farber, S Yuzvinsky, Topological robotics: Subspace arrangements and collision free motion planning (editors V M Buchstaber, I M Krichever), Amer. Math. Soc. Transl. Ser. 2 212, Amer. Math. Soc. (2004) 145 · Zbl 1088.68171
[13] E M Feichtner, G M Ziegler, The integral cohomology algebras of ordered configuration spaces of spheres, Doc. Math. 5 (2000) 115 · Zbl 0992.55014
[14] A M Gleason, Spaces with a compact Lie group of transformations, Proc. Amer. Math. Soc. 1 (1950) 35 · Zbl 0041.36207 · doi:10.2307/2032430
[15] J González, P Landweber, Symmetric topological complexity of projective and lens spaces, Algebr. Geom. Topol. 9 (2009) 473 · Zbl 1167.57012 · doi:10.2140/agt.2009.9.473
[16] N Iwase, M Sakai, Topological complexity is a fibrewise L-S category, Topology Appl. 157 (2010) 10 · Zbl 1192.55002 · doi:10.1016/j.topol.2009.04.056
[17] J W Jaworowski, Extensions of \(G\)-maps and Euclidean \(G\)-retracts, Math. Z. 146 (1976) 143 · Zbl 0301.57025 · doi:10.1007/BF01187702
[18] S Kallel, Symmetric products, duality and homological dimension of configuration spaces (editors N Iwase, T Kohno, R Levi, D Tamaki, J Wu), Geom. Topol. Monogr. 13 (2008) 499 · Zbl 1152.55007 · doi:10.2140/gtm.2008.13.499
[19] R Karasev, P Landweber, Estimating the higher symmetric topological complexity of spheres, Algebr. Geom. Topol. 12 (2012) 75 · Zbl 1244.55013 · doi:10.2140/agt.2012.12.75
[20] J C Latombe, Robot motion planning, Kluwer Int. Series Engin. Comp. Sci. 124, Kluwer Academic (1991) · Zbl 0817.93045
[21] S M LaValle, Planning algorithms, Cambridge Univ. Press (2006) · Zbl 1100.68108 · doi:10.1017/CBO9780511546877
[22] G Lupton, J Scherer, Topological complexity of \(H\)-spaces, Proc. Amer. Math. Soc. 141 (2013) 1827 · Zbl 1263.55002 · doi:10.1090/S0002-9939-2012-11454-6
[23] Y B Rudyak, On higher analogs of topological complexity, Topology Appl. 157 (2010) 916 · Zbl 1187.55001 · doi:10.1016/j.topol.2009.12.007
[24] A S \vSvarc, The genus of a fiber space, Dokl. Akad. Nauk SSSR 119 (1958) 219
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