Involve: A Journal of Mathematics

  • Involve
  • Volume 2, Number 2 (2009), 161-175.

The index of a vector field on an orbifold with boundary

Elliot Paquette and Christopher Seaton

Full-text: Open access

Abstract

A Poincaré–Hopf theorem in the spirit of Pugh is proven for compact orbifolds with boundary. The theorem relates the index sum of a smooth vector field in generic contact with the boundary orbifold to the Euler–Satake characteristic of the orbifold and a boundary term. The boundary term is expressed as a sum of Euler characteristics of tangency and exit-region orbifolds. As a corollary, we express the index sum of the vector field induced on the inertia orbifold to the Euler characteristics of the associated underlying topological spaces.

Article information

Source
Involve, Volume 2, Number 2 (2009), 161-175.

Dates
Received: 11 June 2008
Accepted: 19 February 2009
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1513799138

Digital Object Identifier
doi:10.2140/involve.2009.2.161

Mathematical Reviews number (MathSciNet)
MR2501335

Zentralblatt MATH identifier
1175.57022

Subjects
Primary: 55R91: Equivariant fiber spaces and bundles [See also 19L47] 57R12: Smooth approximations 57R25: Vector fields, frame fields

Keywords
orbifold orbifold with boundary Euler–Satake characteristic Poincare–Hopf theorem vector field vector field index Morse index orbifold double

Citation

Paquette, Elliot; Seaton, Christopher. The index of a vector field on an orbifold with boundary. Involve 2 (2009), no. 2, 161--175. doi:10.2140/involve.2009.2.161. https://projecteuclid.org/euclid.involve/1513799138


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References

  • W. Chen and Y. Ruan, “Orbifold Gromov-Witten theory”, pp. 25–85 in Orbifolds in mathematics and physics (Madison, WI, 2001), Contemp. Math. 310, Amer. Math. Soc., Providence, RI, 2002.
  • W. Chen and Y. Ruan, “A new cohomology theory of orbifold”, Comm. Math. Phys. 248:1 (2004), 1–31.
  • V. Guillemin and A. Pollack, Differential topology, Prentice-Hall, Englewood Cliffs, N.J., 1974.
  • R. Hepworth, “Morse inequalities for orbifold cohomology”, preprint, 2007, http://arxiv.org/pdf/0712.2432.
  • J. Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963.
  • J. W. Milnor, Topology from the differentiable viewpoint, Based on notes by David W. Weaver, The University Press of Virginia, Charlottesville, Va., 1965.
  • I. Moerdijk and D. A. Pronk, “Simplicial cohomology of orbifolds”, Indag. Math. $($N.S.$)$ 10:2 (1999), 269–293.
  • J. R. Munkres, Elementary differential topology, Lectures given at Massachusetts Institute of Technology, Fall 1961, Princeton University Press, Princeton, N.J., 1963.
  • C. C. Pugh, “A generalized Poincaré index formula”, Topology 7 (1968), 217–226.
  • I. Satake, “On a generalization of the notion of manifold”, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 359–363.
  • I. Satake, “The Gauss-Bonnet theorem for $V$-manifolds”, J. Math. Soc. Japan 9 (1957), 464–492.
  • C. Seaton, “Two Gauss-Bonnet and Poincaré-Hopf theorems for orbifolds with boundary”, Differential Geom. Appl. 26:1 (2008), 42–51.
  • W. Thurston, “The geometry and topology of 3-manifolds”, Lecture notes, Princeton University, Mathematics Department, 1978, http://www.msri.org/publications/books/gt3m.
  • S. Waner and Y. Wu, “The local structure of tangent $G$-vector fields”, Topology Appl. 23:2 (1986), 129–143.