Algebraic & Geometric Topology

Morse inequalities for orbifold cohomology

Richard Hepworth

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This paper begins the study of Morse theory for orbifolds, or equivalently for differentiable Deligne–Mumford stacks. The main result is an analogue of the Morse inequalities that relates the orbifold Betti numbers of an almost-complex orbifold to the critical points of a Morse function on the orbifold. We also show that a generic function on an orbifold is Morse. In obtaining these results we develop for differentiable Deligne–Mumford stacks those tools of differential geometry and topology—flows of vector fields, the strong topology—that are essential to the development of Morse theory on manifolds.

Article information

Algebr. Geom. Topol., Volume 9, Number 2 (2009), 1105-1175.

Received: 14 November 2008
Revised: 2 May 2009
Accepted: 7 May 2009
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57N65: Algebraic topology of manifolds 57R70: Critical points and critical submanifolds

Morse theory orbifolds


Hepworth, Richard. Morse inequalities for orbifold cohomology. Algebr. Geom. Topol. 9 (2009), no. 2, 1105--1175. doi:10.2140/agt.2009.9.1105.

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  • D Abramovich, T Graber, A Vistoli, Algebraic orbifold quantum products, from: “Orbifolds in mathematics and physics (Madison, WI, 2001)”, (A Adem, J Morava, Y Ruan, editors), Contemp. Math. 310, Amer. Math. Soc. (2002) 1–24
  • A Adem, J Leida, Y Ruan, Orbifolds and stringy topology, Cambridge Tracts in Math. 171, Cambridge Univ. Press (2007)
  • A Banyaga, D Hurtubise, Lectures on Morse homology, Kluwer Texts in the Math. Sciences 29, Kluwer, Dordrecht (2004)
  • K Behrend, G Ginot, B Noohi, P Xu, String product for inertia stack, Preprint (2006)
  • K Behrend, P Xu, Differentiable stacks and gerbes
  • W Chen, Y Ruan, Orbifold Gromov–Witten theory, from: “Orbifolds in mathematics and physics (Madison, WI, 2001)”, (A Adem, J Morava, Y Ruan, editors), Contemp. Math. 310, Amer. Math. Soc. (2002) 25–85
  • W Chen, Y Ruan, A new cohomology theory of orbifold, Comm. Math. Phys. 248 (2004) 1–31
  • J Heinloth, Notes on differentiable stacks, from: “Mathematisches Institut, Georg-August-Universität Göttingen: Seminars Winter Term 2004/2005”, (Y Tschinkel, editor), Univ. Göttingen (2005) 1–32
  • R Hepworth, Morse theory and the homology of crepant resolutions, in preparation
  • R Hepworth, Orbifold Morse–Smale–Witten theory, in preparation
  • M W Hirsch, Differential topology, Graduate Texts in Math. 33, Springer, New York (1976)
  • D D Joyce, On the topology of desingularizations of Calabi–Yau orbifolds
  • D D Joyce, Compact manifolds with special holonomy, Oxford Math. Monogr., Oxford Univ. Press (2000)
  • S Lang, Differential manifolds, Addison-Wesley, Reading, MA-London-Don Mills, Ont. (1972)
  • E Lerman, S Tolman, Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Amer. Math. Soc. 349 (1997) 4201–4230
  • E Lupercio, B Uribe, M A Xicoténcatl, The loop orbifold of the symmetric product, J. Pure Appl. Algebra 211 (2007) 293–306
  • J Milnor, Morse theory, Annals of Math. Studies 51, Princeton Univ. Press (1963) Based on lecture notes by M Spivak and R Wells
  • I Moerdijk, Orbifolds as groupoids: an introduction, from: “Orbifolds in mathematics and physics (Madison, WI, 2001)”, (A Adem, J Morava, Y Ruan, editors), Contemp. Math. 310, Amer. Math. Soc. (2002) 205–222
  • L I Nicolaescu, An invitation to Morse theory, Universitext, Springer, New York (2007)
  • B Noohi, Foundations of topological stacks I
  • D A Pronk, Etendues and stacks as bicategories of fractions, Compositio Math. 102 (1996) 243–303
  • Y Ruan, The cohomology ring of crepant resolutions of orbifolds, from: “Gromov–Witten theory of spin curves and orbifolds”, (T J Jarvis, T Kimura, A Vaintrob, editors), Contemp. Math. 403, Amer. Math. Soc. (2006) 117–126
  • D Salamon, Morse theory, the Conley index and Floer homology, Bull. London Math. Soc. 22 (1990) 113–140
  • F W Warner, Foundations of differentiable manifolds and Lie groups, Scott, Foresman and Co., Glenview, Ill.-London (1971)
  • A G Wasserman, Equivariant differential topology, Topology 8 (1969) 127–150