Hostname: page-component-55f67697df-jr75m Total loading time: 0 Render date: 2025-05-10T19:48:31.184Z Has data issue: false hasContentIssue false
  • English
  • Français

An n-dimensional Klein bottle

Part of: Classical Topics in Algebraic Topology Homotopy theory Differential topology

Published online by Cambridge University Press:  16 January 2019

Donald M. Davis
Donald M. Davis*
Affiliation:
Department of Mathematics, Lehigh University Bethlehem, PA 18015, USA ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

An n-dimensional analogue of the Klein bottle arose in our study of topological complexity of planar polygon spaces. We determine its integral cohomology algebra and stable homotopy type, and give an explicit immersion and embedding in Euclidean space.

Keywords

Klein bottleimmersionstopological complexitystable homotopy type

MSC classification

Primary: 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirel'man) category of a space
Secondary: 55P15: Classification of homotopy type 57R42: Immersions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 5 , October 2019 , pp. 1207 - 1221
Copyright
Copyright © Royal Society of Edinburgh 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1 Cohen, D. and Vandembroucq, L.. Topological complexity of the Klein bottle, arXiv 1612.03133.Google Scholar
2Colman, H. and Grant, M.. Equivariant topological complexity. Algebr. Geom. Topol. 12 (2012), 2299–2316.Google Scholar
3Cornea, O., Lupton, G., Oprea, J. and Tanré, D.. Lusternik-Schnirelmann category. Math. Surveys Monogr., Amer. Math. Soc. 103 (2003).Google Scholar
4Davis, D. M.. Real projective space as a planar polygon space. Morfismos 19 (2015), 1–6.Google Scholar
5Davis, D. M.. Topological complexity (within 1) of the space of isometry classes of planar n-gons for sufficiently large n. JP J. Geom. Topol. 20 (2017), 1–26.Google Scholar
6Davis, D. M., Explicit motion planning rules in certain polygon spaces, www.lehigh.edu/~dmd1/rules.pdf.Google Scholar
7Farber, M.. Topological complexity of motion planning. Discrete Comput. Geom. 29 (2003), 211–221.Google Scholar
8Farber, M., Tabachnikov, S. and Yuzvinsky, S.. Topological robotics: motion planning in projective spaces. Int. Math. Res. Notes 34 (2003), 1853–1870.Google Scholar
9Franzoni, G., The Klein bottle in its classical shape: a further step towards a good parametrization, arXiv 0909.5354.Google Scholar
10Hausmann, J.-C.. Geometric descriptions of polygon and chain spaces. Contemp. Math. Amer. Math. Soc. 438 (2007), 47–57.Google Scholar
11Hausmann, J.-C. and Knutson, A.. The cohomology rings of polygon spaces. Ann. Inst. Fourier (Grenoble) 48 (1998), 281–321.Google Scholar
12Hirsch, M. W.. Immersion of manifolds. Trans. Amer. Math. Soc. 93 (1959), 242–276.Google Scholar
13Lalonde, F.. Suppression lagrangienne de points doubles et rigidité symplectique. J. Diff. Geom. 36 (1992), 747–764.Google Scholar
14Milnor, J. W. and Stasheff, J. D.. Characteristic classes (Princeton: Annals of Math Studies, 1974).Google Scholar
15Nemirovski, S.. Lagrangian Klein bottles in ℝ2n. Geom. Funct. Anal. 19 (2009), 902–909.Google Scholar