Skagit baker nooksack granite nooksack canyon sumas swift beaver devils rocky n mile eek silesia beaver eek chilliwack lightning slate park fishtrap ruby ruth glacier. Furthermore, we proved that it is generated by two elements of order g gcdg, k for g. Most often it is used in the context of a manifold m. Learn the latest gis technology through free live training seminars, selfpaced courses, or classes taught by esri experts. The mapping class group mods is the group of isotopy classes of homeomorphisms from a surface to itself. In other words, the mapping class group is the group of homeomorphism of x x onto itself, modulo isotopy. The symplectic representation of the mapping class group is. Applications to 3manifolds 11 acknowledgments 12 references 12 1. Presentations of the mapping class group scott morrison. Browse or search thousands of free teacher resources for all grade levels and subjects. In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. So if m is a topological manifold, the mapping class group is the group of isotopyclasses of homeomorphisms of m. An elementary approach to the mapping class group of a. Western regionals 2010 class champions group cd event standings summary 4 races scored.
Lagrangian spheres, symplectic surfaces and the symplectic. We introduce the mapping class group of a surface and its enigmating subgroup, the torelli group. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space. We wish to highlight the connections that are known and point to many interesting open problems in the area. Pdf the stable mapping class group of simply connected 4. One natural problem is to nd a ice presentation of the mapping class group.
The mapping class group is an important algebraic invariant of a surface. We must be very prudent, because there is no manual, no road map, to follow. There exist relatively few equations describing natural phenomena among a great number of partial differential equations pdes. It lies at the juncture of geometry, topology, and group. And while the mapping activity is valuable by itself, the second challenge was and continues to be to compile and. The mapping class group of m is interpreted as the group of isotopyclasses of automorphisms of m.
A theory of random walks on the mapping class group and its nonelementary subgroups is developed. Mapping class groups, homology and finite covers of surfaces. Brendle my research is on braid groups, mapping class groups, and related structures. A projective representation of the mapping class group of a closed surface. Just like wikipedia, you can contribute new information or corrections to the catalog. Lesson plans, unit plans, and classroom resources for your teaching needs. Combining our result with recent work of madsen and weiss, we obtain that the classifying space of the stable mapping class group of nonorientable surfaces, up to homology isomorphism, is the.
The topology of 3manifolds, heegaard distance and the mapping class group of a 2manifold. The mapping class group mapx of such surface x is the group of isotopy classes of orientation preserving homeomorphisms of x which fix pointwise the union of. A presentation for the mapping class group 223 a twodimensional cell complex x, from x, by attaching a 3, 4 or sgon to each cycle of type i, ii or iii above, respectively. Let x be an algebraic curve of genus g, npunctured, defined over a number field k. We prove convergence of sample paths in the thurston compactification and show that the space of projective measured foliations with the corresponding. One hour and seventeen pages are certainly not enough to. The study of mapping class groups, teichmuller geometry and related areas has seen a recent influx of young mathematicians. We prove that the mapping class group of a closed connected orientable surface of genus g is generated by two elements of order g for g. The symplectic representation of the mapping class group is surjective andrew putman abstract we give an e. In mathematics, and more precisely in topology, the mapping class group of a surface, sometimes called the modular group or teichmuller modular group, is the group of homeomorphisms of the surface viewed up to continuous in the compactopen topology deformation. June 9, 2010 abstract we prove that the mapping class group of a.
Humphries and others published generators for the mapping class group find, read and cite all the research you need on researchgate. 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. If x x is a smooth manifold, then the mapping class group is the group of connected components of the diffeomorphism group. Spain lets children play as us states move at various. Portland arts education group for people with disabilities. Open library is an open, editable library catalog, building towards a web page for every book ever published. The asymptotic dimension of mapping class groups is. One hour and seventeen pages are certainly not enough to present this beautiful and rich subject.
Homomorphisms between mapping class groups 1 introduction. Spring isd hosting virtual graduation for class of 2020. Resources are available for professionals, educators, and students. The mapping class group of s, denoted by mods, is the group of isotopy classes of orientationpreserving homeomorphisms of s. So, we recommend for further reading ivanovs survey of the mapping class group 16 as well as farb and.
A primer on mapping class groups benson farb, dan margalit. Problems on mapping class groups and related topics. We characterize the mapping classes that restrict to periodic. We show that if m contains cp2 with either orientation as a connected summand. Then the mapping class group of s, denoted mcgs, is the group of homeomorphisms of sconsidered up to isotopy, where the group operation is composition.
Heres what christian dandrea had to say about highsmith ahead of the draft. It is a vehicle for expressing and releasing the knowledge, creativity, and energy that lies within every group, regardless of its position or level within an organization. Portland arts education group for people with disabilities takes a coronavirus punch, but finds its way with online learning updated apr 21, 2020. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly selfcontained.
While braid groups are technically a special example of a mapping class group, the study. Groups, talk at the 50th cornell topology festival in may 2012 pdf. Mapping class groups are related, both concretely and philosophically, to a number of other groups. The action on homology is given by the well known symplectic representation of the mapping class group. Thus the mapping class group serves as the orbifold fundamental group of the moduli space mg and at the same time it plays the role of the universal mon odromy. Then applying tqft we get an action of the mapping class group on the vector space assigned to the surface. Mapping is merely an enablera means to a more important end. If youre an algebraic geometer, i hope to give you a avour of some simple techniques in symplectic topology. The homology of the mapping class group 3 b a is the tensor product of a polynomial algebra on even dimensional generators and an exterior algebra on odd dimensional generators.
Except for pagination, this version is identical with the published version we have had a longstanding interest in the way that structure in the mapping class group of a. Italys premier giuseppe conte is expected to announce more details on. Athreya, bufetov, eskin, and mirzakhani have shown that the number of orbit points in a teichmuller ball of radius r grows like ehr, where h is the dimension of teichmuller space. Problems on homomorphisms of mapping class groups m. In this article we survey, and make a few new observations about, the surprising connection between submonoids of the mapping class groups and. I will discuss a counting problem for the orbit of the mapping class group in teichmuller space. Cbse syllabus for class 10 social science 20202021. Symplectic topology and algebraic geometry interact in many fruitful ways. Mapping class group notes wade bloomquist these notes look to introduce the mapping class groups of surfaces, explicitly compute examples of low complexity, and introduce the basics on projective representations that will be helpful going forward when looking at quantum representations of mapping class groups.
Pdf the poisson boundary of the mapping class group vadim. On image of tqft representations of mapping class groups. The topology of 3manifolds, heegaard distance and the. It lies at the juncture of geometry, topology, and group theory. Structure of the mapping class groups of surfaces 353 problem 2. Lagrangian spheres, symplectic surfaces and the symplectic mapping class group a dissertation submitted to the faculty of the graduate school of the university of minnesota. Spring isd is planning two graduation ceremonies for class of 2020. The torelli group and representations of mapping class groups. Lectures on the mapping class group of a surface thomas kwokkeung au, feng luo, and tian yang abstract. It is an important algebraic invariant of a surface, and is deeply connected to teichmuller space and the moduli space of riemann surfaces homeomorphic to s.
Track covid19 cases in texas with countybycounty updates. Check the latest cbse syllabus of class 10 social science subject for new academic session 20202021. In chapter 3, we will give some connections between braid groups and mapping class group of the surfaces. Free classroom lesson plans and unit plans for teachers. Given a oriented topological manifold x, its mapping class group. Theorem 2 the mapping class group m2,1 of an orientable surface s2,1 of genus g 2 with one boundary component is generated by. Mapping class groups of heegaard splittings jesse johnson and hyam rubinstein abstract. Project muse mappingclassgroup action versus galois. The contained of this chapter is based on my understanding of section 4. Also we compute the mapping class group of the npunctured sphere. The study of the mapping class group mods is a classical topic that is experiencing a renaissance. The purpose of this thesis is to study the actions of an individual mapping class.
Here i have grouped my papers and book chapters roughly according to topics of particular focus, though there is a bit of overlap. Last name concurrentfirst name university college acton academic statusgregory edge project pe 121 undergraduate individual10. The mapping class group of a heegaard splitting is the group of automorphisms of the ambient 3manifold that take the surface onto itself, modulo isotopies that keep the surface on itself. Let be a complete nitearea orientable hyperbolic surface with one cusp, and let r be the space of complete geodesic rays in emanating from the puncture. Wegiveapreciseconditionintermsofthenielsenthurstondecompositionwhenanelement. Then, the profinite or the prol completion of the topological fundamental group of x admits two actions. Pdf a primer on mapping class groups semantic scholar. Then the mapping class group of s, denoted mcg s, is the group of homeomorphisms of sconsidered up to isotopy, where the group operation is composition. As we hope this survey paper to be highly accessible, insection 2we recall 1 basic. Generators for the mapping class group request pdf. Equivalently it is the group of connected components of aut x \mathbfautx.
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