In these lectures, we give the proofs of two basic theorems on surface topology, namely, the work of dehn and lickorish on generating the mapping class group of a surface by dehntwists. We will prove the ideal class group of every number. Connections with 3manifolds, symplectic geometry and algebraic geometry 129 chapter 9. A primer on mapping class groups begins by explaining the main group theoretical properties of mods, from finite generation by dehn twists and lowdimensional homology to the dehnnielsenbaer theorem. Relation between the braid group and the mapping class group. Database class mapping wizard in pega pega knowledge sharing. If x x is a smooth manifold, then the mapping class group is the group of connected components of the diffeomorphism group. This group is also known as the teichmuller modular group of s, whence the notation. My goal was to present some analytic techniques that can be used to study the action of the mapping class group on the representations varieties of surface groups. A primer on mapping class groups begins by explaining the main grouptheoretical properties of mods, from finite generation by dehn twists and lowdimensional homology to the dehnnielsenbaer theorem.
Journey mapping is a process that provides a holistic view of the customer experience by uncovering moments of both frustration and delight throughout a series of interactions. Mind mapping in testing has become popular in recent years as it assists in lateral. Minsky these notes summarize a short set of lectures i gave at the pcmi program on mapping class groups and moduli spaces in the summer of 2011. Reflection principles and bounds for class group torsion. Remember only concrete classes can be mapped to dedicated database table. The dehnnielsenbaer theorem states that for a closed, connected and orientable surface m the extended mapping class group of m is isomorphic to the outer automorphism group of the fundamental g. Class group calculations keith conrad the minkowski bound says, for a number eld k, that any ideal class contains an integral ideal with norm bounded above by n.
The group of selfhomeomorphisms of a topological space, modulo homotopy. Definition of mapping class group in the dictionary. Mapping class groups and curves in surfaces picture this maths. B of b and we get an embedding of mg into the mapping class group mcgs of the surface s. This is the same as the mapping class group of the torus minus a point x, which acts on loops based at x. Lowdimensional homology groups of mapping class groups. The computation of class numbers, rather than class groups, can be obtained by analytic methods. First, we use the notation in figure 9 with the notation introduced. The mapping class group, m, of m is thegroup of isotopy classes oforientation preserving.
These include the relation between the structure of the mapping class group and invariants of 3manifolds, the unstable cohomology of the moduli space of curves. My job was to introduce the mapping class group of a surface, discuss its basic features from a. In other words, the mapping class group is the group of homeomorphism of x x onto itself, modulo isotopy. Netmap is an interviewbased mapping tool that helps people understand, visualize, discuss, and improve situations in which many different actors influence outcomes. Mapping class group how is mapping class group abbreviated. Along the way, central objects and tools are introduced, such as the birman exact sequence, the complex of curves, the braid group, the. Infinite loop space structures on the stable mapping class. We say that the mapping class group has an isometric action on the curve graph, so to every element of the group we associate an isometry of the graph, which is a map which preserves distances between elements. So if m is a topological manifold, the mapping class group is the group of isotopyclasses of homeomorphisms of m. The factors of z correspond to the dehn twists around the boundary circles of the holes. Now the mapping class group acts on pi well, to be more precise, since a diffeomorphism of the surface might move the basepoint around, you have a homomorphism.
Due to contractual obligations, we will not be posting the published version on the web site. Information and translations of mapping class group in the most comprehensive dictionary definitions resource on the web. Ten problems on the mapping class groups nikolai v. We shall elaborate on arguments from ba for a proof in the general. The least possible cardinality of dehn twists generating the mapping class group of a closed surface of genus g. The group is the mapping class group of s, denoted by mods. Steve humphries showed in 14 that one has the following set of generators for the mapping class group of a surface of genus g. By creating influence network maps, individuals and groups can clarify their own view of a situation, foster discussion, and develop a strategic approach to their networking activities. The torelli group and representations of mapping class groups. We prove that the mapping class group of a closed oriented surface of genus at least two does not have kazhdans property t. Then pi is a free group endowed with a set s of n distinguished conjugacy classes, namely the conjugacy classes of loops around punctures. We will give the complete list of the rst homologies and we calculate them.
In particular, the ideal class group is generated by the prime ideals with norm not exceeding this bound. Pega supports mapping class to the table, so that when we save a page of defined class, the particular instance gets saved to the mapped database table. All the analogies and parallels between mapping class groups and other groups show how mathematics is connected. The ordinary cohomology with rational coefficients of the delooping of the stable mapping class group of 2dimensional manifolds hence essentially the orbifold cohomology of the moduli stack of complex curves is the content of mumford. Abstract we study the large scale geometry of the mapping class group, mcgs. Mapping class groups are discussed m section ll, with an emphasis on the torelli group. These techniques have been around for a while, with origins going back several hundred years. Mapping class groups fall 2017, ken bromberg this course will be an introduction to some basic topics about the mapping class groups with some more advanced topics towards the end. Mapping class groups of punctured surfaces and maybe. As a developer all we need to do is, create a new concrete class. The mapping class group mcg of s is the group of all isotopy classes of orientation preserving selfhomeomorphisms of s. In number theory, the ideal class group or class group of an algebraic number field k is the.
Our main result is that for any asymptotic cone of mcgs, the maximal dimension of locally compact subsets coincides with the maximal. Every homeomorphism h of b induces a homeomorphism of the boundary s. Mapping class groups do not have kazhdans property t. We refer to i02 for a summary of the basic properties of the. The first statement is due to, see also at sphere eversion. Done successfully, it reveals opportunities to address customers pain points, alleviate fragmentation, and, ultimately, create a better experience for your users. Ideal class groups of number fields are fundamental objects in number theory. We will study these two objects and how they relate to each other. 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. Recruiting small group leaders is an ongoing task of the small group leadership team. This resource provides suggestions on key traits to consider when recruiting small group leaders and allows you to map groups by type, age and gender.
Mapping class group factorizations and symplectic 4manifolds 3 gompf,theargumentextendstothecaseoflefschetz. In the case where is a closed orientable surface group, theorem 1. Actions of mapping class groups hal archive ouverte. A brief introduction to mapping class groups yair n. Oneended subgroups of mapping class groups 3 in fact, we could take gto be any gromov hyperbolic space, though we will deal with a graph here to avoid technical details. Ivanov 19941998, 2014 let sbe a compact orientable surface, possibly with a boundary. Pdf it is shown, that the mapping class group of a surface of the genus g 1 admits a faithful representation into the matrix group gl 6g6, z. Minsky dedicated to the memory of candida silveira. Z gg 0,1 of a disk with g holes the boundary of which are parametrised. This proof is based on the presentation of the mapping class group.
Introduction these notes are based on lectures given at zhejiang university, july 1420, 2008. For example, mccarthy and ivanov independently proved that mapping class groups satisfy a version of the tits alternative. The distance between two points in the graph is just the smallest number of edges we need to pass along to get from one to the other. The torelli group and representations of mapping class groups tara e. P consisting of principal divisors that can be generated by a totally positive ele ment. The mapping class groups of all compact orientable 3manifolds are essentially known. Most often it is used in the context of a manifold m. We prove that certain obstructions to the existence of a faithful linear representation do not exist in mg.
An elementary proof of the second homology of the mapping class group was recently given by the author and stipsicz in 30. Mapping class groups and classifying surface automorphisms. Definition of mapping class group in the definitions. The mapping class group of m is interpreted as the group of isotopyclasses of automorphisms of m. Introduction there are at least two interesting actions of the mapping class group modg.
Our book is now available from princeton university press. Lectures on the mapping class group of a surface thomas kwokkeung au, feng luo, and tian yang abstract. Examples include heegaard decompositions of 3manifolds, monodromies of surface bundles, and monodromies of lefschetz. A primer on mapping class groups math the university of utah. A fairly detailed summary of the results, focusing on the nonprime case and with references to proofs in the literature, can be found in section 2 of a paper of mine with nathalie wahl. A linear representation of the mapping class group a and the.
Generators for the mapping class group request pdf. In this group led discussion, marie will demonstrate mind mapping techniques which can be used from test planning through to test closure activities. Whenever the mapping class group of a closed orientable surface of genus gacts by semisimple isometries on a complete cat0 space of dimension less than git xes a point. We refer to 16, 10 for a general introduction to mapping class groups and to 11 for a recent survey on dynamical properties of the action of outg on representation varieties repg. Somewhat less familiar, but equally as important, is the mapping class group of a topological space. From these properties of cut systems it is a routine matter to read off a finite presentation for the mapping class group of m, the group of isotopy classes of. We refer to i02 for a summary of the basic properties of the mapping class group and for references.
Problems on mapping class groups and related topics. It is shown, that the mapping class group of a surface of the genus g 1 admits a faithful representation into the matrix group gl 6g6, z. The proof is based on a categorical correspondence between the riemann surfaces and the socalled toric afalgebras. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. Problems on homomorphisms of mapping class groups m. Pdf the mapping class groups of reducible heegaard. Dimension and rank for mapping class groups by jason a. Mapping class groups and classifying surface automorphisms joseph wells arizona state university november, 2015 the fundamental group of a topological space is a familiar and important algebraic invariant. Nov 01, 2016 we say that the mapping class group has an isometric action on the curve graph, so to every element of the group we associate an isometry of the graph, which is a map which preserves distances between elements.
Equivalently it is the group of connected components of aut x \mathbfautx. Mapping class group factorizations and symplectic 4manifolds. In particular, the mapping class group of a surface is a finitely generated group. In this paper, we survey some rigidity results on actions of mapping class groups. If the class number is prime, then of course the class group is. Stabilization for mapping class groups of 3manifolds, duke math. The dehnnielsenbaer theorem states that for a closed, connected and orientable surface m the extended mapping class group of m is isomorphic to the outer automorphism group of. Class group calculations by minkowski bound keith conrad.
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