Effectual topological complexity. (English) Zbl 1539.55001
In [Discrete Comput. Geom. 29 No. 2, 211–221 (2003; Zbl 1038.68130)], M. Farber introduced the concept of the topological complexity, \(\operatorname{TC}(X)\), of a topological space \(X\), which is the minimal number \(k\) such that there is a partition \(X\times X = E_1\cup\cdots \cup E_k\) admitting a continuous function \(\phi : E_i \to X^I,\,I =[0,1]\) such that \(\phi(x_0, x_1)\) is a path from \(x_0\) to \(x_1\). Later several people suggested different modifications of topological complexity, some of them in the equivariant context.
Given a group \(G\) and a \(G\)-space \(X\), the authors define an invariant \(\operatorname{TC}^G_{\mathrm{effl}}(X)\) as a section category of the fibration \(e: X^I \to X\times X/G\), and call it effectual topological complexity. Here \(e(\gamma)=(\gamma(0), [\gamma(1)])\) and \([-]\) stands for the \(G\)-orbits. Mainly, the authors compare \(\operatorname{TC}^G_{\mathrm{effl}}(X)\) with two other invariants: \(\operatorname{TC}(X/G)\) and the effective \(\operatorname{TC}^G_{\mathrm{effv}}(X)\), see [Z. Błaszczyk and M. Kaluba, Publ. Mat., Barc. 62, No. 1, 55–74 (2018; Zbl 1385.55003)].
Theorem 1.1: If \(X\) is Hausdorff and \(G\) is a discrete group acting properly discontinuously on \(X\), then \[ \operatorname{TC}^G_{\mathrm{effv}}(X)\leq \operatorname{TC}^G_{\mathrm{effl}}(X)\leq \operatorname{TC}(X/G). \] In case of \((X,G)=(T, \mathbb Z/2)\) where \((T,\mathbb Z/2)\) is 2-torus \(T\) with antipodal involution we have the following explicit calculations. Theorem 1.2: \(\operatorname{TC}^{\mathbb Z/2}_{\mathrm{effv}}(T)=2, \, \operatorname{TC}^{\mathbb Z/2}_{\mathrm{effl}}(T)=3,\, \operatorname{TC}(T/(\mathbb Z/2))=4\).
Given a group \(G\) and a \(G\)-space \(X\), the authors define an invariant \(\operatorname{TC}^G_{\mathrm{effl}}(X)\) as a section category of the fibration \(e: X^I \to X\times X/G\), and call it effectual topological complexity. Here \(e(\gamma)=(\gamma(0), [\gamma(1)])\) and \([-]\) stands for the \(G\)-orbits. Mainly, the authors compare \(\operatorname{TC}^G_{\mathrm{effl}}(X)\) with two other invariants: \(\operatorname{TC}(X/G)\) and the effective \(\operatorname{TC}^G_{\mathrm{effv}}(X)\), see [Z. Błaszczyk and M. Kaluba, Publ. Mat., Barc. 62, No. 1, 55–74 (2018; Zbl 1385.55003)].
Theorem 1.1: If \(X\) is Hausdorff and \(G\) is a discrete group acting properly discontinuously on \(X\), then \[ \operatorname{TC}^G_{\mathrm{effv}}(X)\leq \operatorname{TC}^G_{\mathrm{effl}}(X)\leq \operatorname{TC}(X/G). \] In case of \((X,G)=(T, \mathbb Z/2)\) where \((T,\mathbb Z/2)\) is 2-torus \(T\) with antipodal involution we have the following explicit calculations. Theorem 1.2: \(\operatorname{TC}^{\mathbb Z/2}_{\mathrm{effv}}(T)=2, \, \operatorname{TC}^{\mathbb Z/2}_{\mathrm{effl}}(T)=3,\, \operatorname{TC}(T/(\mathbb Z/2))=4\).
Reviewer: Yuli Rudyak (Gainesville)
MSC:
| 55M30 | Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects) |
| 57S25 | Groups acting on specific manifolds |
| 68T40 | Artificial intelligence for robotics |
| 93C85 | Automated systems (robots, etc.) in control theory |
References:
| [1] | Błaszczyk, Z. and Kaluba, M., Effective topological complexity of spaces with symmetries,Publ. Mat.62 (2018) 55-74. · Zbl 1385.55003 |
| [2] | N. Cadavid-Aguilar and J. González, Effective topological complexity of orientable-surface groups, preprint (2020), arXiv:1907.10212v2. |
| [3] | Cohen, D. C. and Vandembroucq, L., Topological complexity of the klein bottle,J. Appl. Comput. Topol.1 (2017) 199-213. · Zbl 1400.55001 |
| [4] | Costa, A. and Farber, M., Motion planning in spaces with small fundamental groups,Commun. Contemp. Math.12 (2010) 107-119. · Zbl 1215.55001 |
| [5] | D. M. Davis, Tables of immersions and embeddings of real projective spaces, available from https://www.lehigh.edu/\( \sim\) dmd1/imms.html. |
| [6] | Dieck, T. T., Algebraic Topology, (Eur. Math. Soc., 2008). · Zbl 1156.55001 |
| [7] | Farber, M., Topological complexity of motion planning,Discrete Comput. Geom.29 (2003) 211-221. · Zbl 1038.68130 |
| [8] | Farber, M., Tabachnikov, S. and Yuzvinsky, S., Topological robotics: Motion planning in projective spaces,Int. Math. Res. Not. IMRN2003 (2003) 1853-1870. · Zbl 1030.68089 |
| [9] | González, J., Grant, M. and Vandembroucq, L., Hopf invariants, topological complexity, and LS-category of the cofiber of the diagonal map for two-cell complexes, in Topological Complexity and Related Topics, , Vol. 702 (Amer. Math. Soc., 2018), pp. 133-150. · Zbl 1388.55003 |
| [10] | Pavešić, P., Topological complexity of a map,Homology Homotopy Appl.21 (2019) 107-130. · Zbl 1426.55004 |
| [11] | Spanier, E. H., Algebraic Topology (McGraw-Hill Book Company, 1966). · Zbl 0145.43303 |
| [12] | Whitehead, G. W., Elements of Homotopy Theory, , Vol. 61 (Springer-Verlag, 1978). · Zbl 0406.55001 |
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