We study orbifolds and strong maps of orbifolds. We begin with introducing a representation for orbifolds that consists of internal categories in the category of topological spaces. These categories are built from atlas charts and chart embeddings without equivalence relation. They represent orbifolds and atlas maps, but do not work well for general strong maps. We generalize the notion of category of fractions to internal categories in the category of topological spaces. We find its universal property for an internal category in the category of topological spaces. We apply this to the atlas category to obtain an atlas groupoid. We give a description of strong maps of orbifolds and the equivalence relation on them in terms of atlas groupoids. We define paths in orbifolds as strong maps. We use our construction to give an explicit description of the equivalence classes on such paths in terms of charts and chart embeddings.

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Dans la première partie de ce mémoire nous introduisons toute la théorie nécessaire à une bonne compréhension de la théorie des variétés et orbifolds toriques. Dans la deuxième partie nous présentons la construction de Delzant d'une variété torique. Finalement, le troisième chapitre présente deux preuves différentes de la formule de Guillemin. ______________________________________________________________________________ MOTS-CLÉS DE L’AUTEUR : Géométrie kählérienne, Orbifold, Variété torique, Polytope de Delzant.

Spectral theory is the study of Mark Kac's famous question [K], "can one hear the shape of a drum?" That is, can we determine the geometrical or topological properties of a manifold by using its Laplace Spectrum? In recent years, the problem has been extended to include the study of Riemannian orbifolds within the same context. In this thesis, on the one hand, we answer Kac's question in the negative for orbifolds that are spherical space forms of dimension higher than eight. On the other hand, for the three-dimensional and four-dimensional cases, we answer Kac's question in the affirmative for orbifold lens spaces, which are spherical space forms with cyclic fundamental groups. We also show that the isotropy types and the topology of the singularities of Riemannian orbifolds are not determined by the Laplace spectrum. This is done in a joint work with E. Stanhope and D. Webb by using P. Berard's generalization of T. Sunada's theorem to obtain isospectral orbifolds. Finally, we construct a technique to get examples of orbifold lens spaces that are not isospectral, but have the same asymptotic expansion of the heat kernel. There are several examples of such pairs in the manifold setting, but to the author's knowledge, the examples developed in this thesis are among the first such examples in the orbifold setting.

The computer programs SnapPea by Weeks and Geo by Casson have proven to be powerful tools in the study of hyperbolic 3-manifolds. Manifolds are special examples of spaces called orbifolds, which are modelled locally on R^n modulo finite groups of symmetries. SnapPea can also be used to study orbifolds but it is restricted to those whose singular set is a link.One goal of this thesis is to lay down the theory for a computer program that can work on a much larger class of 3-orbifolds. The work of Casson is generalized and implemented in a computer program Orb which should provide new insight into hyperbolic 3-orbifolds.The other main focus of this work is the study of 2-handle additions. Given a compact 3-manifold M and an essential simple closed curve α on ∂M, then we define M[α] to be the manifold obtained by gluing a 2-handle to ∂M along α. If α lies on a torus boundary component, we cap off the spherical boundary component created and the result is just Dehn filling.The case when α lies on a boundary surface of genus ≥ 2 is examined and conditions on α guaranteeing that M[α] is hyperbolic are found. This uses a lemma of Scharlemann and Wu, an argument of Lackenby, and a theorem of Marshall and Martin on the density of strip packings. A method for performing 2-handle additions is then described and employed to study two examples in detail.This thesis concludes by illustrating applications of Orb in studying orbifoldsand in the classification of knotted graphs. Hyperbolic invariants are used to distinguish the graphs in Litherland’s table of 90 prime θ-curves and provide access to new topological information including symmetry groups. Then by prescribing cone angles along the edges of knotted graphs, tables of low volume orbifolds are produced.

The Orbifold Landau-Ginzburg Mirror Symmetry Conjecture states that for a quasihomogeneous singularity W and a group G of symmetries of W, there is a dual singularity WT and dual group GT such that the orbifold A-model of W/G is isomorphic to the orbifold B-model of WT/GT. The Landau-Ginzburg A-model is the Frobenius algebra HW,G constructed by Fan, Jarvis, and Ruan, and the B-model is the Orbifold Milnor ring of WT . The unorbifolded conjecture has been verified for Arnol'd's list of simple, unimodal and bimodal quasi-homogeneous singularities with G the maximal diagonal symmetry group by Priddis, Krawitz, Bergin, Acosta, et al. [9], and by Fan-Shen [4] and Acosta [1] for all two dimensional invertible singularities and by Krawitz for all invertible singularities of 3 dimensions and greater in [8]. Based on this Krawitz posed the Orbifold Landau-Ginzburg Mirror Symmetry Conjecture, where the A-model is still the Frobenius algebra HW,G constructed by Fan, Jarvis, and Ruan but constructed with respect to a proper subgroup G of the maximal group of symmetries GW and the B-model is the orbifold Milnor ring of WT orbifolded with respect to a non-trivial group K in SLn of order [GW :J]. I verify this Orbifold Landau-Ginzburg Mirror Symmetry Conjecture for all unimodal and bimodal quasi-homogeneous singularities in Arnol'd's list with G = J < GW, being the minimal admissible diagonal symmetry group. I also discuss some axioms and properties of these singularities.

Nesta tese demonstramos, entre outras coisas, a existência de innitas geodésicas fechadas em good orbifolds Riemannianos M/, onde é um grupo de isometrias virtualmente Abeliano. No caso particular onde é um produto semi-direto de um grupo nito por um grupo Abeliano, concluimos a existência de uma família de geodésicas fechadas com comprimentos tendendo a innito. / In this PhD theses we prove, among other things, the existence of innity many (geometric distinct) closed geodesics on good Riemannian compact orbifolds M/, where is a virtual abelian group of isometries. In the particular case where is a semi-direct product of a nite group with an abelian group, we also assure that there isa family of closed geodesics for which the lengths tend to innity.

No description available.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第19487号 / 理博第4147号 / 新制||理||1596(附属図書館) / 32523 / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 青山 秀明, 教授 田中 貴浩, 教授 畑 浩之 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

In 2002, Giroux showed that every contact structure had a corresponding open book decomposition. This was the converse to a previous construction of Thurston and Winkelnkemper, and made open books a vital tool in the study of contact three-manifolds. We extend these results to contact orbifolds, i.e. spaces that are locally diffeomorphic to the quotient of a contact manifold and a compatible finite group action. This involves adapting some of the main concepts and constructions of three dimensional contact geometry to the orbifold setting.