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Sequential motion planning algorithms in real projective spaces: An approach to their immersion dimension

  • Natalia Cadavid-Aguilar ORCID logo , Jesús González ORCID logo EMAIL logo , Darwin Gutiérrez , Aldo Guzmán-Sáenz and Adriana Lara ORCID logo
Published/Copyright: July 21, 2017
Forum Mathematicum
From the journal Forum Mathematicum Volume 30 Issue 2

Abstract

The s-th higher topological complexity TCs⁡(X) of a space X can be estimated from above by homotopical methods, and from below by homological methods. We give a thorough analysis of the gap between such estimates when X=ℝ⁢Pm, the real projective space of dimension m. In particular, we describe a number r⁢(m), which depends on the structure of zeros and ones in the binary expansion of m, and with the property that 0≤s⁢m-TCs⁡(ℝ⁢Pm)≤δs⁢(m) for s≥r⁢(m), where δs⁢(m)=(0,1,0) for m≡(0,1,2)mod4. Such an estimation for TCs⁡(ℝ⁢Pm) appears to be closely related to the determination of the Euclidean immersion dimension of ℝ⁢Pm. We illustrate the phenomenon in the case m=3⋅2a. In addition, we show that, for large enough s and even m, TCs⁡(ℝ⁢Pm) is characterized as the smallest positive integer t=t⁢(m,s) for which there is a suitable equivariant map from Davis’ projective product space P𝐦s to the (t+1)-st join-power ((ℤ2)s-1)∗(t+1). This is a (partial, but conjecturally complete) generalization of the work of Farber, Tabachnikov and Yuzvinsky relating TC2 to the immersion dimension of real projective spaces.

Keywords: Sectional category; higher topological complexity; zero-divisors cup-length; Euclidean immersion; real projective space; projective product space; Milnor’s join construction
MSC 2010: 55M30; 55R35; 57R42; 68T40; 70B15

Communicated by Frederick R. Cohen

Funding source: Secretaría de Investigación y Posgrado, Instituto Politécnico Nacional

Award Identifier / Grant number: 221221

Funding source: Consejo Nacional de Ciencia y Tecnología

Award Identifier / Grant number: SIP20171446

Funding statement: The second author was partially supported by Conacyt, research grant 221221. The fourth author held a postdoctoral scholarship at Abacus while this work was completed. The fifth author was supported by grant SIP20171446 from Instituto Politécnico Nacional.

Acknowledgements

This work is based on parts of the Ph.D. theses of the first and third authors. As described in Section 1, the first goal of this paper represents the start of the Ph.D. thesis work of Natalia Cadavid-Aguilar, while the second goal of the paper is based on a part of the Ph.D. thesis work of Darwin Gutiérrez. We thank Yuli Rudyak for pointing out the convenience of merging both common-grounded contributions into a single paper. Furthermore, we thank the anonymous referees for their corrections and highly valuable comments.

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. Search in Google Scholar

[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. 10.2140/agt.2014.14.2103Search in Google Scholar

[3] D. C. Cohen and L. Vandembroucq, Topological complexity of the Klein bottle, J. Appl. Comput. Topol. (2017), 10.1007/s41468-017-0002-0. 10.1007/s41468-017-0002-0Search in Google Scholar

[4] A. Costa and M. Farber, Motion planning in spaces with small fundamental groups, Commun. Contemp. Math. 12 (2010), no. 1, 107–119. 10.1142/S0219199710003750Search in Google Scholar

[5] D. M. Davis, A strong nonimmersion theorem for real projective spaces, Ann. of Math. (2) 120 (1984), no. 3, 517–528. 10.2307/1971086Search in Google Scholar

[6] D. M. Davis, Projective product spaces, J. Topol. 3 (2010), no. 2, 265–279. 10.1112/jtopol/jtq006Search in Google Scholar

[7] D. M. Davis, Tables of Euclidean immersions and non-immersions for real projective spaces, http://www.lehigh.edu/~dmd1/imms.html. Search in Google Scholar

[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. 10.1090/proc/13219Search in Google Scholar

[9] M. Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), no. 2, 211–221. 10.1007/s00454-002-0760-9Search in Google Scholar

[10] M. Farber, Instabilities of robot motion, Topology Appl. 140 (2004), no. 2–3, 245–266. 10.1016/j.topol.2003.07.011Search in Google Scholar

[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. 10.1155/S1073792803210035Search in Google Scholar

[12] J. González, M. Grant and L. Vandembroucq, Hopf invariants for sectional category with applications to topological robotics, preprint (2014), https://arxiv.org/abs/1405.6891. 10.1093/qmath/haz019Search in Google Scholar

[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. 10.4310/HHA.2016.v18.n2.a20Search in Google Scholar

[14] N. Kitchloo and W. S. Wilson, The second real Johnson–Wilson theory and nonimmersions of R⁢Pn, Homology Homotopy Appl. 10 (2008), no. 3, 223–268. 10.4310/HHA.2008.v10.n3.a11Search in Google Scholar

[15] G. Lupton and J. Scherer, Topological complexity of H-spaces, Proc. Amer. Math. Soc. 141 (2013), no. 5, 1827–1838. 10.1090/S0002-9939-2012-11454-6Search in Google Scholar

[16] Y. B. Rudyak, On higher analogs of topological complexity, Topology Appl. 157 (2010), no. 5, 916–920. 10.1016/j.topol.2009.12.007Search in Google Scholar

[17] A. Schwarz, The genus of a fiber space, Amer. Math. Soc. Transl. (2) 55 (1966), 49–140. 10.1090/trans2/055/03Search in Google Scholar

Received: 2016-11-17
Revised: 2017-5-17
Published Online: 2017-7-21
Published in Print: 2018-3-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. The variance of divisor sums in arithmetic progressions
  3. Characterizing Lie groups by controlling their zero-dimensional subgroups
  4. CM cycles on Kuga–Sato varieties over Shimura curves and Selmer groups
  5. Towards a Goldberg–Shahidi pairing for classical groups
  6. On the non-existence of cyclic splitting fields for division algebras
  7. Sequential motion planning algorithms in real projective spaces: An approach to their immersion dimension
  8. Very ampleness of the bicanonical line bundle on compact complex 2-ball quotients
  9. Anharmonic solutions to the Riccati equation and elliptic modular functions
  10. A non-homogeneous local Tb theorem for Littlewood–Paley g*λ-function with Lp-testing condition
  11. Saturation rank for finite group schemes: Finite groups and infinitesimal group schemes
  12. On symplectic semifield spreads of PG(5,q2), q odd
  13. From Freudenthal’s spectral theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectra
  14. Golodness and polyhedral products for two-dimensional simplicial complexes
https://doi.org/10.1515/forum-2016-0231
Keywords for this article
Sectional category; higher topological complexity; zero-divisors cup-length; Euclidean immersion; real projective space; projective product space; Milnor’s join construction

Articles in the same Issue

  1. Frontmatter
  2. The variance of divisor sums in arithmetic progressions
  3. Characterizing Lie groups by controlling their zero-dimensional subgroups
  4. CM cycles on Kuga–Sato varieties over Shimura curves and Selmer groups
  5. Towards a Goldberg–Shahidi pairing for classical groups
  6. On the non-existence of cyclic splitting fields for division algebras
  7. Sequential motion planning algorithms in real projective spaces: An approach to their immersion dimension
  8. Very ampleness of the bicanonical line bundle on compact complex 2-ball quotients
  9. Anharmonic solutions to the Riccati equation and elliptic modular functions
  10. A non-homogeneous local Tb theorem for Littlewood–Paley g*λ-function with Lp-testing condition
  11. Saturation rank for finite group schemes: Finite groups and infinitesimal group schemes
  12. On symplectic semifield spreads of PG(5,q2), q odd
  13. From Freudenthal’s spectral theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectra
  14. Golodness and polyhedral products for two-dimensional simplicial complexes
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