Orbifolds and stringy topology department of mathematics. Soren illman, the equivariant triangulation theorem for actions of compact lie groups, math. In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold for orbitmanifold is a generalization of a manifold. Analytical study of different network topologies nivedita bisht1, sapna singh2 1 2assistant professor, e. Stringy geometry of orbifolds indian statistical institute. T pithoragarh, uttarakhand, indiaabstract a network is the interconnection of two or more devices. A base for the topology t is a subcollection t such that for an. Orbifolds lie at the intersection of many different areas of mathematics, includ ing algebraic and differential geometry, topology, algebra, and string theory, among others. Finally, this work also yields some important insights on the nature of orbifolds in string theory. Ruan, orbifolds and stringy topology, cambridge tracts in mathematics 171 cambridge university press, cambridge, 2007. Orbifolds were rst introduced into topology and di erential geometry by satake 6, who called them vmanifolds. Mathematically, such an orbifold cohomology theory has been constructed by chenruan cr.
Sep 22, 2010 there are two approaches to constructing stringy multiplications for global quotients. Calabiyau moduli space, mirror manifolds and spacetime. What is more, although the word orbifold was coined relatively recently,1 orbifolds actually have a much longer history. We define the string homology ring product at the level of the. An introduction to the theory of orbifolds from a modern perspective, combining techniques from geometry, algebraic topology and algebraic geometry. The study of arrangement or mapping of elements links, nodes of a network is known as network topology. This is a survey article on the recent development of stringy geometry and topology of orbifolds, a new subject of mathematics motivated by orbifold string theory. Orbifold string theory model is an popular model in string theory. This is a collection of topology notes compiled by math topology students at the university of michigan in the winter 2007 semester. The first one was the orbifold cohomology of chenruan and then several others appear in the literature such as the stringy product on twisted orbifold ktheory of. Stanford university the homology of the loop space of a manifold has a ring structure that comes from the existence of a pushpull diagram and the existence of an umkher map in homology, the same idea applied in the case of orbifolds, gives the homology.
In these lectures i will begin by recalling the basic structure on the homology of loop spaces and path spaces discovered by chas and sullivan. Find materials for this course in the pages linked along the left. Covering spaces of orbifolds fiberproduct approach pathapproach to the universal covering spaces some helpful references i w. Michael davis, lectures on orbifolds and reflection groups, 2008 pdf. Bus topology is a network type in which every computer and network device is connected to single cable. Standard topology of r let r be the set of all real numbers. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. Questions tagged orbifolds mathematics stack exchange. Thuston, orbifolds and seifert space, chapter 5, notes i a. Index 147 discrete torsion, 76, 98 dolbeault cohomology, 86 dormant sector, 99 effective, 5 groupoid, 19, 46 orbifold, 2 elliptic curve, 8 embedding of charts, 2 groupoids, 35. Stringy topology, cambridge tracts in mathematics 171 2007 pdf. Network topologies michigan technological university.
Even 686 l dixon et al strings on orbifolds though no smooth ricciflat manifolds with o6 holonomy are known 12, we can still consider string theory on oribifolds with broken supersymmetry. Twist fields from orbifold conformaltopological field theory. Alejandro adem, johann leida and yongbin ruan, orbifolds and stringy topology, cambridge tracts in mathematics vol. One can also define orbifold in the topological, that is c0, category. Introduces the theory of orbifolds from a modern perspective, providing a detailed description of the chenruan cohomology. Introduction to topology 5 3 transitivity x yand y zimplies x z.
Discussion of orbifold as lie groupoidsdifferentiable stacks is in. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Ruan, orbifolds and stringy topology, cambridge tracts in mathematics 171 2007 as lie groupoids. Johann leida, yongbin ruan, orbifolds and stringy topology. Stringy geometry and topology of orbifolds internet archive. Orbifolds and stringy topology cambridge tracts in. Download pdf 484 kb abstract this is a survey article on the recent development of stringy geometry and topology of orbifolds, a new subject of mathematics motivated by orbifold string theory. Show that the topological space n of positive numbers with topology generated by arithmetic progression basis is hausdor.
A sampling of these perspectives may be found, for instance, in 1, 11, 32, 48. Cambridge core mathematical physics orbifolds and stringy topology by alejandro adem. Introduction to orbifolds april 25, 2011 1 introduction orbifolds lie at the intersection of many di erent areas of mathematics, including algebraic and di erential geometry, topology, algebra and string theory. An introduction ieke moerdijk 205 orbifold cohomology group of toric varieties mainak poddar 223 hilbert schemes and symmetric products.
Network topology is the schematic description of a network arrangement, connecting various nodes sender and receiver through lines of connection. Ruan, orbifolds and stringy topology, cambridge tracts in mathematics 171 2007. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions. Introduction to topology martina rovelli these notes are an outline of the topics covered in class, and are not substitutive of the lectures, where most proofs are provided and examples are discussed in more detail. Ratcliffe, chapter in foundations of hyperbolic manifolds, springer. One of the main motivations, and a major source of examples, is string theory, where orbifolds play an important role.
Connections to group cohomology and equivariant algebraic topology appear in the context of orbifolds and their. Types of network topology in computer networks studytonight. A quotient orbifold is an orbifold given as the quotient of a smooth, e ective, almost free action of a compact lie group gon a smooth manifold m. Lectures on orbifolds and group cohomology 4 a chart u. Lectures on orbifolds and group cohomology ubc math. Questions tagged orbifolds ask question in the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold for orbitmanifold is a generalization of a manifold. The main topological invariant obtained from orbifold conformal field theory. There are more than two hundred papers on hepth whose title include orbifold. The adjective stringy was coinned by yongbin ruan in order to denote those structures that can be associated to orbifolds which are constructed from loops or strings on the orbifold. In order to apply methods from algebraic topology in the study of orbifolds, we recall a wellknown result about manifolds with smooth actions of compact lie groups see 71. Ruan, orbifolds and stringy topology, cambridge, 2007. Werner ballmann, singular spaces of nonpositive curvature, pages 189201 in sur les groupes hyperboliques dapres mikhael gromov, progress in mathematics 83 1990, birkhauser.
The first one is given by first pulling back and then pushing forward. Today orbifolds are approached from a variety of viewpoints. In pract ice, it may be awkw ard to list all the open sets constituting a topology. One aspect of this new geometry and topology is the twisted orbifold cohomology and its relation to discrete torsion. The second one is given by first pushing forward and then pulling back. The subject is first developed following the classical description analogous to manifold theory, after which the book branches. A dictionary zhenbo qin and weiqiang wang 233 stringy orbifolds yongbin ruan 259 discrete torsion, quotient stacks, and string orbifolds eric sharpe 301 v.
We seek to understand how the geometry and topology of a riemannian orbifold relates to its steklov spectrum. The following observation justi es the terminology basis. Introductory topics of pointset and algebraic topology are covered in a series of five chapters. Its main purpose is to study stringy properties of orbifold, which is unique for orbifold. The first approach has been used to define a global stringy extension of the functors k 0 and k top by jarviskaufmannkimura, a by abramovichgrabervistoli. Lectures on orbifolds and group cohomology alejandro adem and michele klaus abstract. Orbifolds lie at the intersection of many different areas of mathematics, including algebraic and differential geometry, topology, algebra, and string theory, among others. Stringy geometry mathematical congress of the americas 2017. A reference dealing with the string topology of orbifolds is. Stringy topology cuernavaca, mexico, 1618 october 2006. Banyaga, the structure of classical diffeomorphism groups, mathematics and its applications 400 kluwer academic publishers group, dordrecht, 1997. Contemporary mathematics 310 orbifolds in mathematics. The author believes that there is a stringy geometry and topology of orbifolds whose core is orbifold cohomology. In these lectures i will begin by recalling the basic structure on the homology of loop spaces and path spaces discovered by.
In this paper, we introduce the basics of the topology of orbifolds, talk about their. We will discuss these and other matters related to the phenomenology of string theories on orbtfolds more fully in 11. Basic concepts, constructing topologies, connectedness, separation axioms and the hausdorff property, compactness and its relatives, quotient spaces, homotopy, the fundamental group and some application, covering spaces and classification of covering space. The subject is first developed following the classical description analogous. For an element a2xconsider the onesided intervals fb2xja topology. Network topologies topology physical and logical network layout physical actual layout of the computer cables and other network devices logical the way in which the network appears to the devices that use it. Spectral geometry of the steklov problem on orbifolds. It is a topological space called the underlying space with an orbifold structure see below the underlying space locally looks like the quotient space of a euclidean space under the linear action of a finite group. Topological quantum field theories, strings, and orbifolds. Among other topics we will discuss gerbes, discrete torsion and stringy products in the twisted ktheory of orbifolds. We will discuss these and other matters related to the phenomenology of. There are two approaches to constructing stringy multiplications for global quotients.
Orbifoldsstacks stringy orbifoldsstacks motivation. Stringy orbifold ktheory purdue math purdue university. Dorette pronk references 1 alejandro adem, johann leida, yongbin ruan, orbifolds and stringy topology, cambridge tracts in mathematics. The goal of this part of the book is to teach the language of mathematics.
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