Category and topological complexity of the configuration space \(F(G\times \mathbb{R}^n,2)\). (English) Zbl 1428.55006
Let \(F(X,k)\) be the ordered configuration space of \(k\) distinct points, \(F(X,k)=\{(x_1, \ldots, x_k)\mid x_i\neq x_j \text{ for } i\neq j\}\). Let cat and TC denote the non-normalized category and topological complexity, respectively. Let \(zcl\) denote the zero-cup-length. Let \(\mathbb K\) be a field. The main results in the present article are the following.
Theorem 3.9. Let \(G\) be a connected, compact Lie group such that \(\operatorname{cat}(G)=\operatorname{cup}_{\mathbb K}+1\). Then \(\operatorname{cat}(F(G \times \mathbb R^n; 2))=2\operatorname{cat}(G)-1\). Furthermore, \(\operatorname{cat}(F(G \times \mathbb R^n; 2)) = \operatorname{cup}_{\mathbb K}G= (F(G \times \mathbb R^n; 2))+1\).
Theorem 3.15. Let \(G\) be an \(m\)-dimensional connected, compact Lie group such that \(\operatorname{cat}(G \times S^{m+n-1}) =\operatorname{cat}(G)+1\) and \(\mathrm{TC}(G) = \operatorname{zcl}_{\mathbb K}(G)+1\). Then \(\mathrm{TC}(F(G \times \mathbb R^n; 2)) = 2\mathrm{TC}(G) = 2\operatorname{cat}(G)\). Furthermore \(\mathrm{TC}(F(G \times \mathbb R^n; 2)) = \operatorname{zcl}_{\mathbb K}(F(G \times \mathbb R^n; 2))+1\).
Theorem 3.9. Let \(G\) be a connected, compact Lie group such that \(\operatorname{cat}(G)=\operatorname{cup}_{\mathbb K}+1\). Then \(\operatorname{cat}(F(G \times \mathbb R^n; 2))=2\operatorname{cat}(G)-1\). Furthermore, \(\operatorname{cat}(F(G \times \mathbb R^n; 2)) = \operatorname{cup}_{\mathbb K}G= (F(G \times \mathbb R^n; 2))+1\).
Theorem 3.15. Let \(G\) be an \(m\)-dimensional connected, compact Lie group such that \(\operatorname{cat}(G \times S^{m+n-1}) =\operatorname{cat}(G)+1\) and \(\mathrm{TC}(G) = \operatorname{zcl}_{\mathbb K}(G)+1\). Then \(\mathrm{TC}(F(G \times \mathbb R^n; 2)) = 2\mathrm{TC}(G) = 2\operatorname{cat}(G)\). Furthermore \(\mathrm{TC}(F(G \times \mathbb R^n; 2)) = \operatorname{zcl}_{\mathbb K}(F(G \times \mathbb R^n; 2))+1\).
Reviewer: Yuli Rudyak (Gainesville)
MSC:
| 55R80 | Discriminantal varieties and configuration spaces in algebraic topology |
| 55M30 | Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects) |
| 55P10 | Homotopy equivalences in algebraic topology |
Keywords:
configuration spaces; cup-length; zero-divisor cup-length; topological complexity; Lusternik-Schnirelmann categoryReferences:
| [1] | D. C.Cohen, ‘Topological complexity of classical configuration spaces and related objects’, in: Topological Complexity and Related Topics, Contemporary Mathematics, 702 (eds. M.Grant, G.Lupton and L.Vandembroucq) (American Mathematical Society, Providence, RI, 2018), 41-60. · Zbl 1409.55003 |
| [2] | D. C.Cohen and M.Farber, ‘Topological complexity of collision-free motion planning on surfaces’, Compos. Math.147(2) (2011), 649-660. · Zbl 1218.55004 |
| [3] | O.Cornea, G.Lupton, J.Oprea and D.Tanré, Lusternik-Schnirelmann Category, Mathematical Surveys and Monographs, 103 (American Mathematical Society, Providence, RI, 2003). · Zbl 1032.55001 |
| [4] | A.Dranishnikov, ‘Topological complexity of wedges and covering maps’, Proc. Amer. Math. Soc.142(12) (2014), 4365-4376. · Zbl 1305.55003 |
| [5] | A.Dranishnikov and R.Sadykov, ‘The topological complexity of the free product’. Math. Z., to appear. · Zbl 1422.55005 |
| [6] | E.Fadell and L.Neuwirth, ‘Configuration spaces’, Math. Scand.10(4) (1962), 111-118. · Zbl 0136.44104 |
| [7] | M.Farber, ‘Topological complexity of motion planning’, Discrete Comput. Geom.29(2) (2003), 211-221. · Zbl 1038.68130 |
| [8] | M.Farber, ‘Instabilities of robot motion’, Topol. Appl.140(2-3) (2004), 245-266. · Zbl 1106.68107 |
| [9] | M.Farber, Invitation to Topological Robotics (European Mathematical Society, Zurich, 2008). · Zbl 1148.55011 |
| [10] | M.Farber, ‘Configuration spaces and robot motion planning algorithms’, in: Combinatorial and Toric Homotopy: Introductory Lectures (eds. A.Darby, J.Grbić and J.Wu) (World Scientific, Singapore, 2017), 263-303. · Zbl 1402.55002 |
| [11] | M.Farber and M.Grant, ‘Topological complexity of configuration spaces’, Proc. Amer. Math. Soc.137(5) (2009), 1841-1847. · Zbl 1172.55006 |
| [12] | M.Farber, M.Grant and S.Yuzvinsky, ‘Topological complexity of collision free motion planning algorithms in the presence of multiple moving obstacles’, in: Topology Robotics, Contemporary Mathematics, 438 (eds. M.Farber, R.Ghrist, M.Burger and D.Koditschek) (American Mathematical Society, Providence, RI, 2007), 75-84. · Zbl 1143.55300 |
| [13] | M.Farber and S.Yuzvinsky, ‘Topological robotics: subspace arrangements and collision free motion planning’, in: Geometry, Topology, and Mathematical Physics, American Mathematical Society Translations Series 2, 212 (American Mathematical Society, Providence, RI, 2004), 145-156. · Zbl 1088.68171 |
| [14] | J.González and M.Grant, ‘Sequential motion planning of noncolliding particles in Euclidean spaces’, Proc. Amer. Math. Soc.143(10) (2015), 4503-4512. · Zbl 1325.55008 |
| [15] | N.Iwase and M.Sakai, ‘Topological complexity is a fibrewise L-S category’, Topol. Appl.157(1) (2010), 10-21. · Zbl 1192.55002 |
| [16] | N.Iwase, M.Mimura and T.Nishimoto, ‘Lusternik-Schnirelmann category of non-simply connected compact simple Lie groups’, Topol. Appl.150(1-3) (2005), 111-123. · Zbl 1069.55001 |
| [17] | N.Iwase and T.Miyauchi, ‘On Lusternik-Schnirelmann category of SO(10)’, Fund. Math.234(3) (2016), 201-227. · Zbl 1358.55002 |
| [18] | I. M.James, ‘On category, in the sense of Lusternik-Schnirelmann’, Topology17(4) (1978), 331-348. · Zbl 0408.55008 |
| [19] | J.-C.Latombe, Robot Motion Planning (Springer, New York, 1991). |
| [20] | S. M.LaValle, Planning Algorithms (Cambridge University Press, Cambridge, 2006). · Zbl 1100.68108 |
| [21] | F.Roth, ‘On the category of Euclidean configuration spaces and associated fibrations’, Geom. Topol. Monogr.13 (2008), 447-461. · Zbl 1145.55003 |
| [22] | C. A. I.Zapata, ‘Collision-free motion planning on manifolds with boundary’, Preprint, 2017, arXiv:1710.00293. |
| [23] | C. A. I.Zapata, ‘Lusternik-Schnirelmann category of the configuration space of complex projective space’, Topol. Proc.54 (2018), 103-108. · Zbl 1434.55008 |
| [24] | C. A. I.Zapata, ‘Non-contractible configuration spaces’, MORFISMOS22 (2018), 27-39. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
