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Author : Ryan David Budney
Release : 2002
Genre :
Kind : eBook
Book Rating : /5 ( reviews)
Book Synopsis Representation of Mapping Class Groups Via Topological Constructions by : Ryan David Budney
Download or read book Representation of Mapping Class Groups Via Topological Constructions written by Ryan David Budney. This book was released on 2002. Available in PDF, EPUB and Kindle. Book excerpt:
Author : Ryan David Budney
Release : 2002
Genre :
Kind : eBook
Book Rating : 452/5 ( reviews)
Book Synopsis Representations of Mapping Class Groups Via Topological Constructions by : Ryan David Budney
Download or read book Representations of Mapping Class Groups Via Topological Constructions written by Ryan David Budney. This book was released on 2002. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation is an investigation into the action of mapping class groups on objects one would naturally associate with such groups, such as the homology of configuration spaces, homology of covering spaces of configuration spaces and generalized homology theories associated to the underlying surface. The main results of are explicit CW decompositions of configuration spaces, constructed using Morse electrostatic potential functions. In Chapter 3, these are applied to give insight into the Lawrence-Krammer representation, showing that it is a unitary representation and giving some insight into the conjugacy problem for braid groups. Chapter 4 is concerned with the construction of an analogue of the Lawrence-Krammer representation. Chapter 2 is concerned with the action of mapping class groups on the homology of configuration spaces, the main result being that these representations are largely not faithful. In Chapter 5, generalized homology theories are shown to be very similar to standard, singular homology from the point of view of representations of mapping class groups.
Author : Benson Farb
Release : 2006-09-12
Genre : Mathematics
Kind : eBook
Book Rating : 385/5 ( reviews)
Book Synopsis Problems on Mapping Class Groups and Related Topics by : Benson Farb
Download or read book Problems on Mapping Class Groups and Related Topics written by Benson Farb. This book was released on 2006-09-12. Available in PDF, EPUB and Kindle. Book excerpt: The appearance of mapping class groups in mathematics is ubiquitous. The book presents 23 papers containing problems about mapping class groups, the moduli space of Riemann surfaces, Teichmuller geometry, and related areas. Each paper focusses completely on open problems and directions. The problems range in scope from specific computations, to broad programs. The goal is to have a rich source of problems which have been formulated explicitly and accessibly. The book is divided into four parts. Part I contains problems on the combinatorial and (co)homological group-theoretic aspects of mapping class groups, and the way in which these relate to problems in geometry and topology. Part II concentrates on connections with classification problems in 3-manifold theory, the theory of symplectic 4-manifolds, and algebraic geometry. A wide variety of problems, from understanding billiard trajectories to the classification of Kleinian groups, can be reduced to differential and synthetic geometry problems about moduli space. Such problems and connections are discussed in Part III. Mapping class groups are related, both concretely and philosophically, to a number of other groups, such as braid groups, lattices in semisimple Lie groups, and automorphism groups of free groups. Part IV concentrates on problems surrounding these relationships. This book should be of interest to anyone studying geometry, topology, algebraic geometry or infinite groups. It is meant to provide inspiration for everyone from graduate students to senior researchers.
Author : Benson Farb
Release : 2012
Genre : Mathematics
Kind : eBook
Book Rating : 949/5 ( reviews)
Book Synopsis A Primer on Mapping Class Groups by : Benson Farb
Download or read book A Primer on Mapping Class Groups written by Benson Farb. This book was released on 2012. Available in PDF, EPUB and Kindle. Book excerpt: The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. 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 self-contained. The book is suitable for graduate students. A Primer on Mapping Class Groups begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.
Author : Michael Colin Fitzpatrick
Release : 2014
Genre : Class groups (Mathematics)
Kind : eBook
Book Rating : /5 ( reviews)
Book Synopsis Continuous Families of Representations of Mapping Class Groups by : Michael Colin Fitzpatrick
Download or read book Continuous Families of Representations of Mapping Class Groups written by Michael Colin Fitzpatrick. This book was released on 2014. Available in PDF, EPUB and Kindle. Book excerpt: The study of mapping class groups began in the 1920s with Max Dehn and Jakob Nielsen. It was about this time that topology was just being developed, so mapping class groups were of immediate interest, being invariants of topological spaces. The works of Dehn and Nielsen were continued by William Harvey in the 1960s and 70s leading to the development of the curve complex, an important construction still very relevant to mathematics today. William Thurston is another important name in this area since he was able to completely classify homeomorphisms of surfaces in 1976, leading to the famous "Nielsen-Thurston Classification Theorem". Representations were first studied by Carl Gauss in the early 1800s and then explored more thoroughly by Ferdinand Frobenius and Richard Dedekind, among others, at the end of that century. Representation theory has since grown into an extremely important and active area of mathematics today because of its widespread applications to other areas of mathematics and even to other subject areas like physics. Quantum group theory is the youngest area in which this thesis has its roots. This area was formalized and studied extensively for the first time in the 1980s by such mathematicians as Vladimir Drinfeld, Michio Jimbo, and Nicolai Reshetikhin, and immediately found applications in mathematics and theoretical physics. Like representation theory, the study of quantum groups is currently a highly active area of mathematics due to its widespread applications across the mathematical spectrum. In this paper I will present two different methods of constructing projective representations of mapping class groups of surfaces. I will then prove some interesting results concerning each of these methods.
