Braids, Links, and Mapping Class Groups. (AM-82), Volume 82 (eBook)

Joan S. Birman (Autor)

eBook Download: PDF

2016 | 1. Auflage
237 Seiten
Princeton University Press (Verlag)
978-1-4008-8142-0 (ISBN)

Infos

The central theme of this study is Artin's braid group and the many ways that the notion of a braid has proved to be important in low-dimensional topology.In Chapter 1 the author is concerned with the concept of a braid as a group of motions of points in a manifold. She studies structural and algebraic properties of the braid groups of two manifolds, and derives systems of defining relations for the braid groups of the plane and sphere. In Chapter 2 she focuses on the connections between the classical braid group and the classical knot problem. After reviewing basic results she proceeds to an exploration of some possible implications of the Garside and Markov theorems.Chapter 3 offers discussion of matrix representations of the free group and of subgroups of the automorphism group of the free group. These ideas come to a focus in the difficult open question of whether Burau's matrix representation of the braid group is faithful. Chapter 4 is a broad view of recent results on the connections between braid groups and mapping class groups of surfaces. Chapter 5 contains a brief discussion of the theory of "e;plats."e; Research problems are included in an appendix.

Produkt-Details

• Erscheint lt. Verlag: 2.3.2016
• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Graphentheorie
• Schlagworte: Addition • Alexander polynomial • Algebraic Structure • automorphism • Ball (mathematics) • bijection • braid group • Braid theory • Branched covering • Burau representation • Calculation • Cartesian coordinate system • Characterization (mathematics) • coefficient • Combinatorial group theory • commutative property • Commutator subgroup • Configuration space • conjugacy class • corollary • Covering space • Dehn twist • Determinant • Diagram (category theory) • Dimension • Disjoint union • Double coset • Eigenvalues and Eigenvectors • Enumeration • Equation • equivalence class • exact sequence • existential quantification • Faithful Representation • Finite set • free abelian group • free group • fundamental group • Geometry • Group (mathematics) • Groupoid • group ring • Handlebody • Heegaard splitting • Homeomorphism • Homomorphism • Homotopy • Homotopy group • Identity element • Identity matrix • Inclusion map • Initial point • Integer • Integer matrix • Knot polynomial • knot theory • Lens space • Line–line intersection • Line segment • Link group • Low-dimensional topology • Mapping class group • Mathematical Induction • Mathematics • matrix group • matrix representation • Monograph • Morphism • Natural transformation • Normal matrix • Notation • Orientability • Parity (mathematics) • Permutation • Piecewise linear • Pointwise • polynomial • Prime knot • Projection (mathematics) • Proportionality (mathematics) • Quotient Group • Requirement • Rewriting • Riemann surface • Semigroup • Sequence • Special case • SUBGROUP • submanifold • Subset • Symmetric group • Theorem • theory • Topology • Trefoil knot • two-dimensional space • Unimodular matrix • Unit vector • Variable (mathematics) • Word problem (mathematics)
• ISBN-10: 1-4008-8142-0 / 1400881420
• ISBN-13: 978-1-4008-8142-0 / 9781400881420