Abstract
We show that the normalized topological complexity of the Klein bottle is equal to 4. For any non-orientable surface \(N_g\) of genus \(g\ge 2\), we also show that \({{{\mathrm{\mathsf{TC}}}}}(N_g)=4\). This completes recent work of Dranishnikov on the topological complexity of non-orientable surfaces.
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References
Birman, J.: Braids, Links and Mapping Class Groups. Annals of Mathematics Studies, vol. 82. Princeton University Press, Princeton (1974) (MR0375281)
Brown, K.S.: Cohomology of Groups, Graduate Texts in Mathematics, vol. 87. Springer, New York (1982) (MR0672956)
Cohen, D., Suciu, A.: Homology of iterated semidirect products of free groups. J. Pure Appl. Algebra 126, 87–120 (1998) (MR1600518)
Cohen, D., Vandembroucq, L.: On the cofibre of the diagonal map of the Klein bottle, (in preparation)
Costa, A., Farber, M.: Motion planning in spaces with small fundamental groups. Commun. Contemp. Math. 12, 107–119 (2010) (MR2649230)
Davis, D.: An approach to the topological complexity of the Klein bottle. Preprint arXiv:1612.02747v5 (2017)
Dranishnikov, A.: The topological complexity and the homotopy cofiber of the diagonal map for non-orientable surfaces. Proc. Am. Math. Soc. 144, 4999–5014 (2016) (MR3544546)
Dranishnikov, A.: On topological complexity of non-orientable surfaces, Topol. Appl. Special Issue dedicated to Kodama, to appear
Farber, M.: Topological complexity of motion planning. Discrete Comput. Geom. 29, 211–221 (2003) (MR1957228)
Farber, M.: Invitation to Topological Robotics. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2008) (MR2455573)
Farber, M., Tabachnikov, S., Yuzvinsky, S.: Topological robotics: motion planning in projective spaces. Int. Math. Res. Not. 34, 1853–1870 (2003) (MR1988783)
Fox, R.: Free differential calculus I. Ann. Math. 57, 547–560 (1953) (MR0053938)
Fox, R.: Free differential calculus II. Ann. Math. 59, 196–210 (1954) (MR0062125)
Fox, R.: Free differential calculus III. Ann. Math. 64, 407–419 (1956) (MR0095876)
Gonçalves, D., Martins, S.: Diagonal approximation and the cohomology ring of the fundamental groups of surfaces. Eur. J. Math. 1, 122–137 (2015) (MR3386228)
González, J., Gutiérrez, B., Gutiérrez, D., Lara, A.: Motion planning in real flag manifolds. Homol. Homot. Appl. 18, 359–375 (2016) (MR3576004)
Rudyak, Y.: On higher analogs of topological complexity. Topol. Appl. 157, 916–920 (2010) (MR2593704)
Schwarz, A.: The genus of a fiber space. Am. Math. Soc. Trans. 55, 49–140 (1966) (MR0154284)
Acknowledgements
D.C. partially supported by NSF 1105439. L.V. partially supported by Portuguese Funds through FCT—Fundação para a Ciência e a Tecnologia, within the Project UID/MAT/00013/2013. We thank Michael Farber for a number of valuable discussions concerning the theme of this article, Jesús González and Mark Grant for many helpful suggestions on our first version, the anonymous referees for useful comments, and the Mathematisches Forschungsinstitut Oberwolfach, where this collaboration started during the mini-workshop Topological Complexity and Related Topics.
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Cohen, D.C., Vandembroucq, L. Topological complexity of the Klein bottle. J Appl. and Comput. Topology 1, 199–213 (2017). https://doi.org/10.1007/s41468-017-0002-0
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DOI: https://doi.org/10.1007/s41468-017-0002-0