Sequential motion planning algorithms in real projective spaces: an approach to their immersion dimension. (English) Zbl 1395.55005
Let \(\text{TC}_s(X)\) be the \(s\)th higher (also sequential) topological complexity (\(\text{TC}\)) of \(X\), due to Y. B. Rudyak [Topology Appl. 157, 916–920 (2010; Zbl 1187.55001)]. This is a natural generalization of Farber’s \(\text{TC}_2(X)\); see, for instance, [M. Farber, Discrete Comput. Geom. 29, 211–221 (2003; Zbl 1038.68130)]. The authors give a detailed analysis of the gap between homotopical methods from above and homological methods from below for real projective \(m\)-space, \(X=\mathbb R P^m\).
By cohomological methods, the authors prove, for instance: the inequalities \(0\leq \delta_s(m) \leq 2^{e(m)} -1\) hold provided \(s\geq \ell(m)\). Here \(\ell(m)=\max\{(m+1)/2^{e(m)},2\}\) and \(e(m)\) stands for the length of the block of consecutive ones ending the binary expansion of \(m\) (e.g., \(e(m) = 0\) if \(m\) is even). In particular, \(\delta_s(m)=0\), so that \(\text{TC}_s(\mathbb RP^m) = sm\), if \(m\) is even and \(s > m\).
The first goal of this paper is to show that a large portion of the initial elements in the sequence \({\delta_{\ell(m)}}(m), {\delta_{\ell(m)-1}}(m), \dots, {\delta_2}(m)\) remain well controlled, in the sense that they satisfy the inequality \(\delta_s(m) \leq 2^{e(m)} -1\).
The second aim is to present an alternative characterization for each of the numbers in the monotonic sequence \(\dots \geq \text{TC}_s(\mathbb RP^m)\geq \text{TC}_{s-1}(\mathbb RP^m)\geq \dots \geq \text{TC}_{2}(\mathbb RP^m)\).
The results of the article are analysed (in Section 7) with the perspective of the Euclidean immersion problem for \(\mathbb R P^m\).
By cohomological methods, the authors prove, for instance: the inequalities \(0\leq \delta_s(m) \leq 2^{e(m)} -1\) hold provided \(s\geq \ell(m)\). Here \(\ell(m)=\max\{(m+1)/2^{e(m)},2\}\) and \(e(m)\) stands for the length of the block of consecutive ones ending the binary expansion of \(m\) (e.g., \(e(m) = 0\) if \(m\) is even). In particular, \(\delta_s(m)=0\), so that \(\text{TC}_s(\mathbb RP^m) = sm\), if \(m\) is even and \(s > m\).
The first goal of this paper is to show that a large portion of the initial elements in the sequence \({\delta_{\ell(m)}}(m), {\delta_{\ell(m)-1}}(m), \dots, {\delta_2}(m)\) remain well controlled, in the sense that they satisfy the inequality \(\delta_s(m) \leq 2^{e(m)} -1\).
The second aim is to present an alternative characterization for each of the numbers in the monotonic sequence \(\dots \geq \text{TC}_s(\mathbb RP^m)\geq \text{TC}_{s-1}(\mathbb RP^m)\geq \dots \geq \text{TC}_{2}(\mathbb RP^m)\).
The results of the article are analysed (in Section 7) with the perspective of the Euclidean immersion problem for \(\mathbb R P^m\).
Reviewer: Július Korbaš (Bratislava)
MSC:
| 55M30 | Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects) |
| 68T40 | Artificial intelligence for robotics |
| 57R42 | Immersions in differential topology |
| 55R35 | Classifying spaces of groups and \(H\)-spaces in algebraic topology |
| 70B15 | Kinematics of mechanisms and robots |
References:
| [1] | J. Adem, S. Gitler and I. M. James, On axial maps of a certain type, Bol. Soc. Mat. Mexicana (2) 17 (1972), 59-62.; Adem, J.; Gitler, S.; James, I. M., On axial maps of a certain type, Bol. Soc. Mat. Mexicana (2), 17, 59-62 (1972) · Zbl 0311.57016 |
| [2] | I. Basabe, J. González, Y. B. Rudyak and D. Tamaki Higher topological complexity and its symmetrization, Algebr. Geom. Topol. 14 (2014), no. 4, 2103-2124.; Basabe, I.; González, J.; Rudyak, Y. B.; Tamaki, D., Higher topological complexity and its symmetrization, Algebr. Geom. Topol., 14, 4, 2103-2124 (2014) · Zbl 1348.55005 |
| [3] | D. C. Cohen and L. Vandembroucq, Topological complexity of the Klein bottle, J. Appl. Comput. Topol. (2017), 10.1007/s41468-017-0002-0.; Cohen, D. C.; Vandembroucq, L., Topological complexity of the Klein bottle, J. Appl. Comput. Topol. (2017) · Zbl 1400.55001 · doi:10.1007/s41468-017-0002-0 |
| [4] | A. Costa and M. Farber, Motion planning in spaces with small fundamental groups, Commun. Contemp. Math. 12 (2010), no. 1, 107-119.; Costa, A.; Farber, M., Motion planning in spaces with small fundamental groups, Commun. Contemp. Math., 12, 1, 107-119 (2010) · Zbl 1215.55001 |
| [5] | D. M. Davis, A strong nonimmersion theorem for real projective spaces, Ann. of Math. (2) 120 (1984), no. 3, 517-528.; Davis, D. M., A strong nonimmersion theorem for real projective spaces, Ann. of Math. (2), 120, 3, 517-528 (1984) · Zbl 0574.57014 |
| [6] | D. M. Davis, Projective product spaces, J. Topol. 3 (2010), no. 2, 265-279.; Davis, D. M., Projective product spaces, J. Topol., 3, 2, 265-279 (2010) · Zbl 1198.55008 |
| [7] | D. M. Davis, Tables of Euclidean immersions and non-immersions for real projective spaces, .; <element-citation publication-type=”other“> Davis, D. M.Tables of Euclidean immersions and non-immersions for real projective spacesPreprint <ext-link ext-link-type=”uri“ xlink.href=”>http://www.lehigh.edu/ dmd1/imms.html |
| [8] | A. Dranishnikov, The topological complexity and the homotopy cofiber of the diagonal map for non-orientable surfaces, Proc. Amer. Math. Soc. 144 (2016), no. 11, 4999-5014.; Dranishnikov, A., The topological complexity and the homotopy cofiber of the diagonal map for non-orientable surfaces, Proc. Amer. Math. Soc., 144, 11, 4999-5014 (2016) · Zbl 1352.55002 |
| [9] | M. Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), no. 2, 211-221.; Farber, M., Topological complexity of motion planning, Discrete Comput. Geom., 29, 2, 211-221 (2003) · Zbl 1038.68130 |
| [10] | M. Farber, Instabilities of robot motion, Topology Appl. 140 (2004), no. 2-3, 245-266.; Farber, M., Instabilities of robot motion, Topology Appl., 140, 2-3, 245-266 (2004) · Zbl 1106.68107 |
| [11] | M. Farber, S. Tabachnikov and S. Yuzvinsky, Topological robotics: Motion planning in projective spaces, Int. Math. Res. Not. IMRN 2003 (2003), no. 34, 1853-1870.; Farber, M.; Tabachnikov, S.; Yuzvinsky, S., Topological robotics: Motion planning in projective spaces, Int. Math. Res. Not. IMRN, 2003, 34, 1853-1870 (2003) · Zbl 1030.68089 |
| [12] | J. González, M. Grant and L. Vandembroucq, Hopf invariants for sectional category with applications to topological robotics, preprint (2014), .; <element-citation publication-type=”other“> González, J.Grant, M.Vandembroucq, L.Hopf invariants for sectional category with applications to topological roboticsPreprint2014 <ext-link ext-link-type=”uri“ xlink.href=”>https://arxiv.org/abs/1405.6891 · Zbl 1388.55003 |
| [13] | J. González, B. Gutiérrez, D. Gutiérrez and A. Lara, Motion planning in real flag manifolds, Homology Homotopy Appl. 18 (2016), no. 2, 359-275.; González, J.; Gutiérrez, B.; Gutiérrez, D.; Lara, A., Motion planning in real flag manifolds, Homology Homotopy Appl., 18, 2, 359-275 (2016) · Zbl 1359.55003 |
| [14] | N. Kitchloo and W. S. Wilson, The second real Johnson-Wilson theory and nonimmersions of RP^n, Homology Homotopy Appl. 10 (2008), no. 3, 223-268.; Kitchloo, N.; Wilson, W. S., The second real Johnson-Wilson theory and nonimmersions of RP^n, Homology Homotopy Appl., 10, 3, 223-268 (2008) · Zbl 1160.55002 |
| [15] | G. Lupton and J. Scherer, Topological complexity of H-spaces, Proc. Amer. Math. Soc. 141 (2013), no. 5, 1827-1838.; Lupton, G.; Scherer, J., Topological complexity of H-spaces, Proc. Amer. Math. Soc., 141, 5, 1827-1838 (2013) · Zbl 1263.55002 |
| [16] | Y. B. Rudyak, On higher analogs of topological complexity, Topology Appl. 157 (2010), no. 5, 916-920.; Rudyak, Y. B., On higher analogs of topological complexity, Topology Appl., 157, 5, 916-920 (2010) · Zbl 1187.55001 |
| [17] | A. Schwarz, The genus of a fiber space, Amer. Math. Soc. Transl. (2) 55 (1966), 49-140.; Schwarz, A., The genus of a fiber space, Amer. Math. Soc. Transl. (2), 55, 49-140 (1966) |
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.
