Invariants Of Homology 3 Spheres
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| Author | : Nikolai Saveliev |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 229 |
| Release | : 2013-04-17 |
| Genre | : Mathematics |
| ISBN | : 3662047055 |
The book gives a systematic exposition of the diverse ideas and methods in the area, from algebraic topology of manifolds to invariants arising from quantum field theories. The main topics covered include: constructions and classification of homology 3-spheres, Rokhlin invariant, Casson invariant and its extensions, and Floer homology and gauge-theoretical invariants of homology cobordism. Many of the topics covered in the book appear in monograph form for the first time. The book gives a rather broad overview of ideas and methods and provides a comprehensive bibliography. The text will be a valuable source for both the graduate student and researcher in mathematics and theoretical physics.
| Author | : Nikolai Saveliev |
| Publisher | : Walter de Gruyter |
| Total Pages | : 220 |
| Release | : 1999 |
| Genre | : Mathematics |
| ISBN | : 9783110162721 |
| Author | : V. M. Buchstaber |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 204 |
| Release | : 1997 |
| Genre | : Geometry |
| ISBN | : 9780821806661 |
| Author | : Tomotada Ohtsuki |
| Publisher | : World Scientific |
| Total Pages | : 516 |
| Release | : 2002 |
| Genre | : Invariants |
| ISBN | : 9789812811172 |
This book provides an extensive and self-contained presentation of quantum and related invariants of knots and 3-manifolds. Polynomial invariants of knots, such as the Jones and Alexander polynomials, are constructed as quantum invariants, i.e. invariants derived from representations of quantum groups and from the monodromy of solutions to the Knizhnik-Zamolodchikov equation. With the introduction of the Kontsevich invariant and the theory of Vassiliev invariants, the quantum invariants become well-organized. Quantum and perturbative invariants, the LMO invariant, and finite type invariants of 3-manifolds are discussed. The ChernOCoSimons field theory and the WessOCoZuminoOCoWitten model are described as the physical background of the invariants. Contents: Knots and Polynomial Invariants; Braids and Representations of the Braid Groups; Operator Invariants of Tangles via Sliced Diagrams; Ribbon Hopf Algebras and Invariants of Links; Monodromy Representations of the Braid Groups Derived from the KnizhnikOCoZamolodchikov Equation; The Kontsevich Invariant; Vassiliev Invariants; Quantum Invariants of 3-Manifolds; Perturbative Invariants of Knots and 3-Manifolds; The LMO Invariant; Finite Type Invariants of Integral Homology 3-Spheres. Readership: Researchers, lecturers and graduate students in geometry, topology and mathematical physics."
| Author | : Robert Lipshitz |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 294 |
| Release | : 2018-08-09 |
| Genre | : Mathematics |
| ISBN | : 1470428881 |
The authors construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes in two different versions, one of which (type D) is a module over the algebra and the other of which (type A) is an A∞ module. Both are well-defined up to chain homotopy equivalence. For a decomposition of a 3-manifold into two pieces, the A∞ tensor product of the type D module of one piece and the type A module from the other piece is ^HF of the glued manifold. As a special case of the construction, the authors specialize to the case of three-manifolds with torus boundary. This case can be used to give another proof of the surgery exact triangle for ^HF. The authors relate the bordered Floer homology of a three-manifold with torus boundary with the knot Floer homology of a filling.
| Author | : S. K. Donaldson |
| Publisher | : Cambridge University Press |
| Total Pages | : 277 |
| Release | : 1990 |
| Genre | : Mathematics |
| ISBN | : 0521399785 |
Distinguished researchers reveal the way different subjects (topology, differential and algebraic geometry and mathematical physics) interact in a text based on LMS Durham Symposium Lectures.
| Author | : Hanna Nencka |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 266 |
| Release | : 1999 |
| Genre | : Mathematics |
| ISBN | : 0821808842 |
"The book has two main parts. The first is devoted to the Poincare conjecture, characterizations of PL-manifolds, covering quadratic forms of links and to categories in low dimensional topology that appear in connection with conformal and quantum field theory.
| Author | : Louis H. Kauffman |
| Publisher | : World Scientific |
| Total Pages | : 578 |
| Release | : 2012 |
| Genre | : Mathematics |
| ISBN | : 9814307998 |
More recently, Khovanov introduced link homology as a generalization of the Jones polynomial to homology of chain complexes and Ozsvath and Szabo developed Heegaard-Floer homology, that lifts the Alexander polynomial. These two significantly different theories are closely related and the dependencies are the object of intensive study. These ideas mark the beginning of a new era in knot theory that includes relationships with four-dimensional problems and the creation of new forms of algebraic topology relevant to knot theory. The theory of skein modules is an older development also having its roots in Jones discovery. Another significant and related development is the theory of virtual knots originated independently by Kauffman and by Goussarov Polyak and Viro in the '90s. All these topics and their relationships are the subject of the survey papers in this book.
| Author | : Nikolai Saveliev |
| Publisher | : Walter de Gruyter |
| Total Pages | : 220 |
| Release | : 2011-12-23 |
| Genre | : Mathematics |
| ISBN | : 3110250365 |
Progress in low-dimensional topology has been very quick in the last three decades, leading to the solutions of many difficult problems. Among the earlier highlights of this period was Casson's λ-invariant that was instrumental in proving the vanishing of the Rohlin invariant of homotopy 3-spheres. The proof of the three-dimensional Poincaré conjecture has rendered this application moot but hardly made Casson's contribution less relevant: in fact, a lot of modern day topology, including a multitude of Floer homology theories, can be traced back to his λ-invariant. The principal goal of this book, now in its second revised edition, remains providing an introduction to the low-dimensional topology and Casson's theory; it also reaches out, when appropriate, to more recent research topics. The book covers some classical material, such as Heegaard splittings, Dehn surgery, and invariants of knots and links. It then proceeds through the Kirby calculus and Rohlin's theorem to Casson's invariant and its applications, and concludes with a brief overview of recent developments. The book will be accessible to graduate students in mathematics and theoretical physics familiar with some elementary algebraic and differential topology, including the fundamental group, basic homology theory, transversality, and Poincaré duality on manifolds.
| Author | : José Seade |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 243 |
| Release | : 2006-03-21 |
| Genre | : Mathematics |
| ISBN | : 3764373954 |
Offers an overview of selected topics on the topology of singularities, with emphasis on its relations to other branches of geometry and topology. This book studies real analytic singularities which arise from the topological and geometric study of holomorphic vector fields and foliations.
