Morse inequalities for orbifold cohomology.

*(English)*Zbl 1175.55004The 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.

Reviewer: Dorin Andrica (Cluj-Napoca)

##### 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 |

##### Keywords:

coarse moduli spaces; differentiable stack; differentiable Deligne-Mumford stacks; Morse function; Morse Lemma; Riemannian metric; vector field; Morse inequalities
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\textit{R. Hepworth}, Algebr. Geom. Topol. 9, No. 2, 1105--1175 (2009; Zbl 1175.55004)

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