Osaka Journal of Mathematics

Generalised spin structures on 2-dimensional orbifolds

Hansjörg Geiges and Jesús Gonzalo Pérez

Full-text: Open access


Generalised spin structures, or $r$-spin structures, on a $2$-dimensional orbifold $\Sigma$ are $r$-fold fibrewise connected coverings (also called $r$\textsuperscript{th} roots) of its unit tangent bundle $ST\Sigma$. We investigate such structures on hyperbolic orbifolds. The conditions on $r$ for such structures to exist are given. The action of the diffeomorphism group of $\Sigma$ on the set of $r$-spin structures is described, and we determine the number of orbits under this action and their size. These results are then applied to describe the moduli space of taut contact circles on left-quotients of the $3$-dimensional geometry $\widetilde{\mathrm{SL}}_{2}$.

Article information

Osaka J. Math., Volume 49, Number 2 (2012), 449-470.

First available in Project Euclid: 20 June 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds 57R18: Topology and geometry of orbifolds 53C27: Spin and Spin$^c$ geometry 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx]


Geiges, Hansjörg; Gonzalo Pérez, Jesús. Generalised spin structures on 2-dimensional orbifolds. Osaka J. Math. 49 (2012), no. 2, 449--470.

Export citation


  • L. Alvarez-Gaumé, G. Moore and C. Vafa: Theta functions, modular invariance, and strings, Comm. Math. Phys. 106 (1986), 1–40.
  • M.F. Atiyah: Riemann surfaces and spin structures, Ann. Sci. École Norm. Sup. (4) 4 (1971), 47–62.
  • F.A. Belgun, N. Ginoux and H.-B. Rademacher: A singularity theorem for twistor spinors, Ann. Inst. Fourier (Grenoble) 57 (2007), 1135–1159.
  • L. Dąbrowski and R. Percacci: Spinors and diffeomorphisms, Comm. Math. Phys. 106 (1986), 691–704.
  • C. Dong, K. Liu and X. Ma: On orbifold elliptic genus; in Orbifolds in Mathematics and Physics (Madison, WI, 2001), Contemp. Math. 310, Amer. Math. Soc., Providence, RI, 87–105, 2002.
  • H. Geiges and J. Gonzalo: Contact geometry and complex surfaces, Invent. Math. 121 (1995), 147–209.
  • \begingroup H. Geiges and J. Gonzalo: Seifert invariants of left-quotients of $3$-dimensional simple Lie groups, Topology Appl. 66 (1995), 117–127. \endgroup
  • H. Geiges and J. Gonzalo: Moduli of contact circles, J. Reine Angew. Math. 551 (2002), 41–85.
  • T.J. Jarvis: Geometry of the moduli of higher spin curves, Internat. J. Math. 11 (2000), 637–663.
  • D. Johnson: Spin structures and quadratic forms on surfaces, J. London Math. Soc. (2) 22 (1980), 365–373.
  • M. Klukas: Engelstrukturen, Diplomarbeit, Universität zu Köln, 2008.
  • R. Kulkarni, K.B. Lee and F. Raymond: Deformation spaces for Seifert manifolds; in Geometry and Topology (College Park, Md., 1983/84), Lecture Notes in Math. 1167, Springer, Berlin, 180–216, 1985.
  • H.B. Lawson, Jr. and M.-L. Michelsohn: Spin Geometry, Princeton Mathematical Series 38, Princeton Univ. Press, Princeton, NJ, 1989.
  • W.D. Neumann and F. Raymond: Seifert manifolds, plumbing, $\mu$-invariant and orientation reversing maps; in Algebraic and Geometric Topology (Proc. Sympos., Univ. California, Santa Barbara, Calif., 1977), Lecture Notes in Math. 664, Springer, Berlin, 163–196, 1978.
  • A. Polishchuk: Witten's top Chern class on the moduli space of higher spin curves; in Frobenius Manifolds (Bonn, 2002), Aspects Math. E36, Vieweg, Wiesbaden, 253–264, 2004.
  • F. Raymond and A.T. Vasquez: $3$-manifolds whose universal coverings are Lie groups, Topology Appl. 12 (1981), 161–179.
  • P. Scott: The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983), 401–487.
  • P.L. Sipe: Some finite quotients of the mapping class group of a surface, Proc. Amer. Math. Soc. 97 (1986), 515–524.
  • N. Steenrod: The Topology of Fibre Bundles, Princeton Mathematical Series 14, Princeton Univ. Press, Princeton, NJ, 1951.
  • E. Witten: Algebraic geometry associated with matrix models of two-dimensional gravity; in Topological Methods in Modern Mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 235–269, 1993.
  • H. Zieschang: On the homeotopy group of surfaces, Math. Ann. 206 (1973), 1–21.