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Morse inequalities for orbifold cohomology. (English) Zbl 1175.55004
The main result of this long and interesting paper is to prove an analogue of the Morse inequalities relating the orbifold Betti numbers of an almost-complex orbifold to the critical points of a Morse function on the orbifold. The paper is organized as follows. In Section 2, the author introduces the underlying space of a differentiable stack. Section 3 reviews the definition of differentiable Deligne-Mumford stack, its relation with proper étale groupoids and orbifolds, and some basic properties such as paracompactness, partition of unity and the existence of orbifold-charts. Section 4 is devoted to the study of Morse functions on differentiable Deligne-Mumford stacks and to the proof of a Morse Lemma describing the local form of a Morse function around a critical point. In Section 5, Riemannian metrics, vector fields, the gradient vector field and integrals and flows of vector fields on differentiable Deligne-Mumford stacks are presented. Section 6 defines the strong topology on the set of real-valued morphisms, extending in this way the classical situation for the algebra of real smooth mappings of a manifold. In Section 7, the author proves the main result of the paper (Theorem 7.11), i.e. the Morse inequalities for each of the three notions of homology or cohomology of an orbifold. Section 8 presents three examples of Morse functions and the related consequences to the Morse inequalities. The first two demonstrate how the main result allows us to compute the homology of Chen-Ruan cohomology groups of an orbifold. The third example shows how the method can be exended to compute the integer homology groups of the K3 surface.

MSC:
55N32 Orbifold cohomology
57R18 Topology and geometry of orbifolds
57N65 Algebraic topology of manifolds
57R70 Critical points and critical submanifolds in differential topology
58E99 Variational problems in infinite-dimensional spaces
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[1] D Abramovich, T Graber, A Vistoli, Algebraic orbifold quantum products (editors A Adem, J Morava, Y Ruan), Contemp. Math. 310, Amer. Math. Soc. (2002) 1 · Zbl 1067.14055
[2] A Adem, J Leida, Y Ruan, Orbifolds and stringy topology, Cambridge Tracts in Math. 171, Cambridge Univ. Press (2007) · Zbl 1157.57001
[3] A Banyaga, D Hurtubise, Lectures on Morse homology, Kluwer Texts in the Math. Sciences 29, Kluwer (2004) · Zbl 1080.57001
[4] K Behrend, G Ginot, B Noohi, P Xu, String product for inertia stack, Preprint (2006) · Zbl 1253.55007
[5] K Behrend, P Xu, Differentiable stacks and gerbes · Zbl 1227.14007
[6] W Chen, Y Ruan, Orbifold Gromov-Witten theory (editors A Adem, J Morava, Y Ruan), Contemp. Math. 310, Amer. Math. Soc. (2002) 25 · Zbl 1091.53058
[7] W Chen, Y Ruan, A new cohomology theory of orbifold, Comm. Math. Phys. 248 (2004) 1 · Zbl 1063.53091
[8] J Heinloth, Notes on differentiable stacksat Göttingen: Seminars Winter Term 2004/2005” (editor Y Tschinkel), Univ. Göttingen (2005) 1 · Zbl 1098.14501
[9] R Hepworth, Morse theory and the homology of crepant resolutions, in preparation
[10] R Hepworth, Orbifold Morse-Smale-Witten theory, in preparation
[11] M W Hirsch, Differential topology, Graduate Texts in Math. 33, Springer (1976) · Zbl 0356.57001
[12] D D Joyce, On the topology of desingularizations of Calabi-Yau orbifolds
[13] D D Joyce, Compact manifolds with special holonomy, Oxford Math. Monogr., Oxford Univ. Press (2000) · Zbl 1027.53052
[14] S Lang, Differential manifolds, Addison-Wesley (1972) · Zbl 0239.58001
[15] E Lerman, S Tolman, Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Amer. Math. Soc. 349 (1997) 4201 · Zbl 0897.58016
[16] E Lupercio, B Uribe, M A Xicoténcatl, The loop orbifold of the symmetric product, J. Pure Appl. Algebra 211 (2007) 293 · Zbl 1148.55005
[17] J Milnor, Morse theory, Annals of Math. Studies 51, Princeton Univ. Press (1963)
[18] I Moerdijk, Orbifolds as groupoids: an introduction (editors A Adem, J Morava, Y Ruan), Contemp. Math. 310, Amer. Math. Soc. (2002) 205 · Zbl 1041.58009
[19] L I Nicolaescu, An invitation to Morse theory, Universitext, Springer (2007) · Zbl 1131.57002
[20] B Noohi, Foundations of topological stacks I · Zbl 1052.14001
[21] D A Pronk, Etendues and stacks as bicategories of fractions, Compositio Math. 102 (1996) 243 · Zbl 0871.18003
[22] Y Ruan, The cohomology ring of crepant resolutions of orbifolds (editors T J Jarvis, T Kimura, A Vaintrob), Contemp. Math. 403, Amer. Math. Soc. (2006) 117 · Zbl 1105.14078
[23] D Salamon, Morse theory, the Conley index and Floer homology, Bull. London Math. Soc. 22 (1990) 113 · Zbl 0709.58011
[24] F W Warner, Foundations of differentiable manifolds and Lie groups, Scott, Foresman and Co. (1971) · Zbl 0241.58001
[25] A G Wasserman, Equivariant differential topology, Topology 8 (1969) 127 · Zbl 0215.24702
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