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In February 2001, scientists at the Department of Energy's Los Alamos National Laboratory announced that they had recorded a simple knot untying itself. Crafted from a chain of nickel-plated steel balls connected by thin metal rods, the three-crossing knot stretched, wiggled, and bent its way out of its predicament--a neat trick worthy of an inorganic Houdini, but more than that, a critical discovery in how granular and filamentary materials such as strands of DNA and polymers entangle and enfold themselves.
A knot seems a simple, everyday thing, at least to anyone who wears laced shoes or uses a corded telephone. In the mathematical discipline known as topology, however, knots are anything but simple: at 16 crossings of a "closed curve in space that does not intersect itself anywhere," a knot can take one of 1,388,705 permutations, and more are possible. All this thrills mathematics professor Colin Adams, whose primer offers an engaging if challenging introduction to the mysterious, often unproven, but, he suggests, ultimately knowable nature of knots of all kinds--whether nontrivial, satellite, torus, cable, or hyperbolic. As perhaps befits its subject, Adams's prose is sometimes, well, tangled ("a knot is amphicheiral if it can be deformed through space to the knot obtained by changing every crossing in the projection of the knot to the opposite crossing"), but his book is great fun for puzzle and magic buffs, and a useful reference for students of knot theory and other aspects of higher mathematics. --Gregory McNamee
Book Description
Over a century old, knot theory is today one of the most active areas of modern mathematics. The study of knots has led to important applications in DNA research and the synthesis of new molecules, and has had a significant impact on statistical mechanics and quantum field theory. Colin Adams's The Knot Book is the first book to make cutting-edge research in knot theory accessible to a non-specialist audience. Starting with the simplest knots, Adams guides readers through increasingly more intricate twists and turns of knot theory, exploring problems and theorems mathematicians can now solve, as well as those that remain open. He also explores how knot theory is providing important insights in biology, chemistry, physics, and other fields. The new paperback edition has been updated to include the latest research results, and includes hundreds of illustrations of knots, as well as worked examples, exercises and problems. With a simple piece of string, an elementary mathematical background, and The Knot Book, anyone can start learning about some of the most advanced ideas in contemporary mathematics.
Customer Reviews:
Good Introduction to Knots.......2007-07-03
In terms of content, I would rate this book 4-5 stars. However, I rated it three stars because it had a flaw in terms of readibility. If you are willing to devote a lot of time to the subject, and are willing to take the time to work through all of the exercises, then this is the book for you. However, if you are just looking for some light reading on an unusual subject, there is a problem with the book. In many cases, if you don't complete the exercises, your ability to understand what follows in the chapter will be impaired. I bought the book to read on the train, and did not really have the facilities to work through all of the exercises. For me, the book would be greatly improved if solutions to some of the exercises (at least in sketch form) were included as an appendix.
In addition to being a good introduction to knots, the book also covers many othet topics in topology as well. At the end of the book, the author tries to show that there are practical applications to knot theory, but for the most part he appeared to be stretching. It seems that knot theory is pretty close to being "pure" mathematics. One thing that he did miss, however, was the application of knot theory to tying neckties. That would have been really practical!
Written for a non-mathematician but certainly enjoyable by mathematicians!.......2006-11-16
This book is aimed at making knot theory accessible to people with little mathematical background, and it does so beautifully. However, the material is not watered down--and there is quite a lot of material in this book, as well as a number of open questions (which are quite difficult). The book starts with basics and seems easy, but it gets into challenging concepts rather quickly. Knot theory is one area of abstract mathematics that is particularly accessible to people with little background and this book works off this assumption quite well. Most importantly, this book is fun--it brings out the fun in the subject, and in mathematics in general!
This book would make excellent reading for anyone who likes puzzles, abstract thought, or novel forms of mathematics. It also would be interesting for mathematicians who want an introduction to knot theory. Someone who wants a more mathematical (but still accessible) treatment might want to check out "Knots and Surfaces" by N. D. Gilbert. In some respects it is a natural follow-up to this book. It is slightly more concise and has more rigorous mathematics in it.
Pretty good introduction.......2005-03-31
One can make nothing wrong buying this book. It gives an easy introduction, and most parts are well explained. Don't expect to become an expert in knot theory after reading it but at least you are then familiar with the basics.
Great introduction to knot theory.......2003-03-09
Having first been exposed to interesting knots while in undergraduate courses in biology and chemistry and occasionally encountering knots in my mathematical life, I have long maintained a passing interest in the field. However, until now, no single event evoked a reaction strong enough to pique a desire to explore. All it took to change that was the reading of this book by Adams.
Surprisingly complete for an introductory text, it is also amazingly understandable. Requiring only knowledge of polynomials and a mind capable of understanding twists, I found it addictive. This is one area where it pays not to think straight. After reading it twice, I still pick it up and scan it in odd moments. Problems are scattered throughout the book, and many can be solved using only a piece of string. Those that are still unsolved are clearly marked, with is good, since the statements are often very simple.
There are many applications and the number is growing all the time. One of the most profound images and statements of discovery was the pictures of the knotting of the rings of Saturn and commentator Carl Sagan saying, "We don't understand that at all. We will have to invent a whole new branch of physics to understand it." The most esoteric recent explanation of the structure of the universe is the theory of superstrings, where all objects are multi-dimensional knots. A fascinating problem in molecular biology of the gene is the process whereby DNA coils when quiescent and uncoils to be copied. One chapter is devoted to applications, although more would have been helpful.
A non-convoluted introduction to the theory of convolutions, this book belongs in every mathematical library.
Published in Journal of Recreational Mathematics, reprinted with permission.
Excellent motivation for knot theory.......2002-06-27
Knot theory has been a branch of mathematics that has been around for over a century, and now is finding applications in mnay areas, some of these being electrical circuit analysis, genetics, dynamical systems, and cryptography. This book, written for the layman or the beginning student of mathematics, is an excellent overview of what is known (and not known) in knot theory. Because of the pictorial nature of the subject, knot theory is an excellent way to get people interested in mathematics. Knot theory now is an established branch of mathematics, and it involves the use of tools from topology, analysis, and algebra. The problem of distinguishing one knot from another is one of the major questions in knot theory, and its partial resolution has been assisted by concepts from physics, namely statistical mechanics and quantum field theory. The author discusses the knot recognition problem, and other unsolved problems in the book, and he points out that in knot theory the unsolved problems can be approached by someone with very little background in advanced mathematical techniques. The author does an excellent job of introducing these problems and letting the reader experience, in his words, the joy of doing mathematics.
Chapter 1 is an introduction to the basic terminology of knot theory, and the author gives examples of the most popular elementary knots. He points out the historical origins of the theory, one of these being the attempt by Lord Kelvin to explain the origins of the elements, interestingly. The basic operations on knots are defined, such as composition and factoring, and the famous Reidemeister moves. The proof that planar isotopies and Reidemeister moves suffices to map one projection of a knot to another is omitted. After defining links and linking numbers, the author then discusses tricolorability, and uses this to prove that there are nontrivial knots.
Chapter 2 then overviews the strategies used in the tabulation of knots.The Dowker notation, used to describe a projection of a knot, is discussed as a tool for listing knots with 13 or less crossings. The author also discusses the Conway notation, and how it is used to study tangles and mutants. Graph theory is also introduced as a technique to study knot projections. The author discusses the unsolved problem of finding an elementary integer function that gives the prime knots with given crossing number, a problem that has important ramifications for cryptography (but the author does not discuss this application).
Since knots are complicated objects, then like many other areas in topology, the strategy is to assign a quantity to a knot that will distinguish it from all other knots. Such a quantity is called an invariant, and as one might guess, no one has yet found an invariant to distinguish all nontrivial knots from each other. In the last two decades though, new powerful knot invariants have been discovered, many of these being based on concepts from theoretical physics. In chapter 3, the author discusses the unknotting number, the bridge number, and the crossing number as elementary examples of knot invariants.
Chapter 4 is more complicated, in that the author shows how to use surfaces to assist in the understanding of knots. After discussing how to triangulate an surface and the concept of a homoeomorphism between surfaces, he introduces the Euler characteristic as an invariant of surfaces. Surfaces appear in knot theory as the space in the knot's complement, and the author introduces the concept of the compressibility of a surface, also very important in three-dimensional topology. Particular attention is paid to Seifert surfaces, which, given a particular knot, are orientable surfaces with one boundary component such that the boundary component is the knot in question.
Several different types of knots are considered in chapter 5, such as torus, satellite and hyperbolic knots. The latter are particularly interesting, since their study is part of the field of hyperbolic geometry, a subject that is now undergoing intense study. The author also introduces the theory of braids and the braid group. Not only are braids very important in the study of knots, but they have taken on major importance in cryptography and dynamical systems.
Chapter 6 is very interesting, and introduces some of the more contemporary topics in knot theory. The assignment of polynomials to knots goes back to the early 20th century, but it took the work of Vaughan Jones and his use of ideas from operator theory and statistical mechanics to provide polynomial invariants of knots that were much finer than the Alexander polynomial of the 1930s. The Jones polynomial however is not introduced the way Jones did, but instead via the Kaufmann bracket polynomial. The HOMFLY polynomial is introduced as a polynomial that generalizes the Jones and Alexander polynomials.
A few applications of knot theory are discussed in Chapter 7, such as the DNA molecule and topological stereoisomers. The author also discusses the applications of knot theory to the theory of exactly solvable models in statistical mechanics, a topic that has mushroomed in the past decade. This is followed by a brief overview of applications of knot theory to graph theory in chapter 8.
Chapters 9 and 10 are an introduction to knot theory as it relates to research in the topology of 3-dimensional manifolds and the existence of knots in dimensions higher than 3. The concepts introduced, particulary the idea of a Heegaard diagram, are used extensively in the study of 3-manifolds. In addition, the author mentions the famous Poincare conjecture, albeit in non-rigorous terms. The Kirby calculus, which is a kind of generalization of the Reidemeister moves, but instead models the sequence of operations that allow one to change from one Dehn surgery description of a 3-manifold to another is briefly discussed. The author also gives a few elementary, intuitive hints about how to visualize knotted objects in high dimensions.
Book Description
This book offers a systematic and comprehensive presentation of the concepts of a spin manifold, spinor fields, Dirac operators, and A-genera, which, over the last two decades, have come to play a significant role in many areas of modern mathematics. Since the deeper applications of these ideas require various general forms of the Atiyah-Singer Index Theorem, the theorems and their proofs, together with all prerequisite material, are examined here in detail. The exposition is richly embroidered with examples and applications to a wide spectrum of problems in differential geometry, topology, and mathematical physics. The authors consistently use Clifford algebras and their representations in this exposition. Clifford multiplication and Dirac operator identities are even used in place of the standard tensor calculus. This unique approach unifies all the standard elliptic operators in geometry and brings fresh insights into curvature calculations. The fundamental relationships of Clifford modules to such topics as the theory of Lie groups, K-theory, KR-theory, and Bott Periodicity also receive careful consideration. A special feature of this book is the development of the theory of Cl-linear elliptic operators and the associated index theorem, which connects certain subtle spin-corbordism invariants to classical questions in geometry and has led to some of the most profound relations known between the curvature and topology of manifolds.
Customer Reviews:
Excellent.......2001-12-22
Who would have known that the equation discovered by P.A.M. Dirac in the 1920's would have the enormous appllications to mathematics that it currently has. This book is an excellent overview of these applications, written by two individuals who are responsible for the development of many of these. Dirac's theory of course had its origins in physics, and physicists, particularly those working in high energy physics, will find this book interesting and helpful.
The authors give a brief introduction and then move on to the representation theory of Clifford algebras and spin groups in chapter 1. The reader can see the origin of Clifford algebras and an introduction to the Pin and Spin groups. Clifford algebras are classified as matrix algebras over the real or complex numbers, and the quaternions. It is the representation theory of Clifford algebras however that has resulted in the impressive results outlined in the book Noting that the tensor product of Clifford algebras is not necessarily a Clifford algebra, the authors introduce a Z(2)-grading on a Clifford algebra, which results in a multiplicative structure in the representations of Clifford algebras. The Lie algebras of the Pin and Spin groups are discussed along with applications to geometry and Lie groups. By far the most interesting discussion though is on K-theory, which allows one to define a ring structure on vector bundles. Distinguishing a base point in the base space, relative K-groups are defined, and shown to be equal for the base space and its i-fold suspension. Bott periodicity results are stated but their proof is delayed until chapter 3. A detailed discussion is given of the Atiyah-Bott-Shapiro isomorphism and KR-theory.
The connection between spin and differential geometry is discussed in chapter 2. The first few sections is a review of standard results in the spin structure of vector bundles, such as Stiefel-Whitney classes and spin cobordism. For Riemannian vector bundles, each fiber has a quadratic form that gives rise to a Clifford algebra on the fiber. The question as to when a vector bundle over the Riemannian base space can be found that has fibers each an irreducible module over this Clifford algebra leads to a consideration of spin manifolds and spin cobordism, when the total space is chosen to be the tangent bundle. The Dirac operator acting on a bundle over this Clifford bundle allows the construction of all the standard elliptic operators such as the signature, Atiyah-Singer, and the Euler characteristic. The authors discuss these constructions in detail along with the notion of of Cl(k)-linear operators.
The Dirac operator can be viewed in Euclidean space as the square root of a Laplace operator, but over general manifolds it is the Laplacian with a correction term dependent on the curvature and Clifford multiplication. The Bochner vanishing theorems are discussed in great detail, along with the results on the existence of exotic spheres.
An entire chapter is spent on index theorems, wherein the authors present the results in terms of the approach used by Atiyah and Singer, instead of the heat kernel methods of Gilkey and Patodi. Physicists might prefer the later approach, due to its connections with applications, but the abstract K-theory approach undertaken by the authors is elegant and their presentation is excellent. The role of physics in index theorems is a fascinating one though, especially the use of supersymmetry to simplify the proofs of some of the results. The authors do not discuss this approach, but point out, interestingly, that it does not work when one is dealing with torsion elements in K-theory. These cannot be detected using cohomology nor can the modulo-two invariants appearing in the index theorems be computed from local densities.
The last chapter is a long one and discusses applications in differential topology and geometry, emphasizing index thoerems and Riemannian manifolds of positive scalar curvature. The authors outline just when the indexes are integers (the integrality theorems) and use spin geometry to discuss the immersion problem for manifolds and the vector field problem. Exotic n-spheres again make their appearance, wherein it is shown that some of these have very few symmetries and are very asymmetric objects. A short introduction to elliptic genera is given. Interestingly, C*-algebras are briefly mentioned as tools to decide whether for every compact spin manifold with positive scalar curvature all higher A-genera must be zero. Spin-c manifolds are not treated, the authors instead concentrating their attention to Kahlerian geometry. In this context the Clifford algebra multiplication has a beautiful relationship with the complex structure. A brief discussion is given of the pure spinors of Cartan and twistor spaces. The theory of holonomy and calibrations, the later due to one of the authors, is discussed in great detail. The discussion begins in the consideration of when universal covering spaces are not Riemannian manifolds and their holonomy groups have been classified. The idea of a calibration arises from the consideration of submanifolds that are homologically volume-minimizing. These become calibrations when the integrals of p-forms on them are the volumes, and these p-forms have vanishing differentials on oriented tangent p-planes on the manifold. The authors give an interesting discussion of the relation between spinors and calibrations.
Essential for grad students in geometry/topology.......1998-12-23
As a graduate student in mathematics I survived on this encyclopedic work. Anyone interested in differential geometry or differential topology will eventually need something in this book.
Prerequisites are graduate-level algebra and analysis, and some topology and differential geometry. He introduces the subject of pseudodifferential operators and Sobolev spaces, but it's easy to get lost in that part unless you first read Shubin's book "Pseudodifferential operators and Spectral theory". Also, the quick shuffling of Lie group information can be disheartening if you're not used to it. Harvey's book "Spinors and Calibrations" is a more elementary book if this is the case.
This book touches on many important topics like the Atiyah-Singer Index Theorem, the Bochner method, Riemann-Roch, and mathematical physics, but you will probably want to supplement your reading with individual books on each of these topics.
Book Description
Since the times of Gauss, Riemann, and Poincaré, one of the principal goals of the study of manifolds has been to relate local analytic properties of a manifold with its global topological properties. Among the high points on this route are the Gauss-Bonnet formula, the de Rham complex, and the Hodge theorem; these results show, in particular, that the central tool in reaching the main goal of global analysis is the theory of differential forms.
The book by Morita is a comprehensive introduction to differential forms. It begins with a quick introduction to the notion of differentiable manifolds and then develops basic properties of differential forms as well as fundamental results concerning them, such as the de Rham and Frobenius theorems. The second half of the book is devoted to more advanced material, including Laplacians and harmonic forms on manifolds, the concepts of vector bundles and fiber bundles, and the theory of characteristic classes. Among the less traditional topics treated is a detailed description of the Chern-Weil theory.
The book can serve as a textbook for undergraduate students and for graduate students in geometry.
Customer Reviews:
Self contained introduction to techniques of classifying manifolds........2007-01-10
This text is phenomenally easy to read and well organized. The author starts you on a journey by first explaining the importance and power of classifying manifolds namely by certain invariants preserved by certain mappings ( diffeomorphisms ).
For example, like Euler, we could count the number of holes in the surface and using this combinatorial method we are led to homology theory.
Or like Gauss, we could use a differentiation and integration to come up with the idea of curvature as an intrinsic feature of the surface.
Modern approaches use differential forms to represent homology and cohomoly groups.
The author also deals with fibre bundles demonstrating their importance in analyzing manifolds specifically how the number of fibre bundles possible with given Lie groups as structure groups over the manifold can be answered by characteristic classes such as the Chern and Pontrjagin classes. The use of differential forms is indispensible.
Perhaps the most satisfying aspect of this book is that it clarifies the notions of connection, connection form, curvature, curvature form for manifolds and fibre bundles.
There are plenty of exercises to boot.
A very good book........2005-03-28
This is probably the most clearly written self-contained book on the basics of differential geometry. The author does a great job explaining the ideas behind purely mathematical 'dry' constructions. On the other hand, everything is defined correctly and precisely. A very readable and useful book with the perfect combination of formal math. and intuition. I would recommend it to students in theoretical physics area, together with the Nakahara's fantastic book.
Book Description
This book provides the first lucid and accessible account to the modern study of the geometry of four-manifolds. It has become required reading for postgraduates and research workers whose research touches on this topic. Pre-requisites are a firm grounding in differential topology, and geometry as may be gained from the first year of a graduate course. The subject matter of this book is the most significant breakthrough in mathematics of the last fifty years, and Professor Donaldson won a Fields medal for his work in the area. The authors start from the standpoint that the fundamental group and intersection form of a four-manifold provides information about its homology and characteristic classes, but little of its differential topology. It turns out that the classification up to diffeomorphism of four-manifolds is very different from the classification of unimodular forms and that the study of this question leads naturally to the new Donaldson invariants of four-manifolds. A central theme of this book is that the appropriate geometrical tools for investigating these questions come from mathematical physics: the Yang-Mills theory and anti-self dual connections over four-manifolds. One of the many consquences of this theory is that 'exotic' smooth manifolds exist which are homeomorphic but not diffeomorphic to (4, and that large classes of forms cannot be realized as intersection forms whereas distinct manifolds may share the same form. These result have had far-reaching consequences in algebraic geometry, topology, and mathematical physics, and will continue to be a mainspring of mathematical research for years to come.
Customer Reviews:
An excellent summary of Donaldson theory.......2000-06-16
This book brings together the brilliant work Donaldson did at Oxford during the early 1980s. The unique properties of 4-manifolds are clearly and concisely written out with concentration on explaining field theories like Yang-Mills and gauge theory with a truly firm mathematical foundation, presented in a book for the first time. A great companion for any researcher in the field of geometry and topology, or even loop quantum gravity!
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Braids and Self-Distributivity (Progress in Mathematics)
Patrick Dehornoy
Manufacturer: Birkhäuser Basel
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ASIN: 3764363436 |
Book Description
This is the award-winning monograph of the Sunyer i Balaguer Prize 1999.
The aim of this book is to present recently discovered connections between Artin’s braid groups and left self-distributive systems, which are sets equipped with a binary operation satisfying the identity x(yz) = (xy)(xz). Order properties are crucial.
In the 1980s new examples of left self-distributive systems were discovered using unprovable axioms of set theory, and purely algebraic statements were deduced. The quest for elementary proofs of these statements led to a general theory of self-distributivity centered on a certain group that captures the geometrical properties of this identity. This group happens to be closely connected with Artin’s braid groups, and new properties of the braids naturally arose as an application, in particular the existence of a left invariant linear order, which subsequently received alternative topological constructions.
The text proposes a first synthesis of this area of research. Three domains are considered here, namely braids, self-distributive systems, and set theory. Although not a comprehensive course on these subjects, the exposition is self-contained, and a number of basic results are established. In particular, the first chapters include a rather complete algebraic study of Artin’s braid groups.
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Abstract Regular Polytopes
Peter McMullen , and
Egon Schulte
Manufacturer: Cambridge University Press
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ASIN: 0521814960 |
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Abstract regular polytopes stand at the end of more than two millennia of geometrical research, which began with regular polygons and polyhedra. The rapid development of the subject in the past twenty years has resulted in a rich new theory featuring an attractive interplay of mathematical areas, including geometry, combinatorics, group theory and topology. This is the first comprehensive, up-to-date account of the subject and its ramifications. It meets a critical need for such a text, because no book has been published in this area since Coxeter's "Regular Polytopes" (1948) and "Regular Complex Polytopes" (1974).
Download Description
Abstract regular polytopes stand at the end of more than two millennia of geometrical research, which began with regular polygons and polyhedra. They are highly symmetric combinatorial structures with distinctive geometric, algebraic or topological properties; in many ways more fascinating than traditional regular polytopes and tessellations. The rapid development of the subject in the past 20 years has resulted in a rich new theory, featuring an attractive interplay of mathematical areas, including geometry, combinatorics, group theory and topology. Abstract regular polytopes and their groups provide an appealing new approach to understanding geometric and combinatorial symmetry. This is the first comprehensive up-to-date account of the subject and its ramifications, and meets a critical need for such a text, because no book has been published in this area of classical and modern discrete geometry since Coxeter's Regular Polytopes (1948) and Regular Complex Polytopes (1974). The book should be of interest to researchers and graduate students in discrete geometry, combinatorics and group theory.
Book Description
Spectral sequences are among the most elegant and powerful methods of computation in mathematics. This book describes some of the most important examples of spectral sequences and some of their most spectacular applications. The first part treats the algebraic foundations for this sort of homological algebra, starting from informal calculations. The heart of the text is an exposition of the classical examples from homotopy theory, with chapters on the Leray-Serre spectral sequence, the Eilenberg-Moore spectral sequence, the Adams spectral sequence, and, in this new edition, the Bockstein spectral sequence. The last part of the book treats applications throughout mathematics, including the theory of knots and links, algebraic geometry, differential geometry and algebra. This is an excellent reference for students and researchers in geometry, topology, and algebra.
Customer Reviews:
A superb overview .......2004-09-13
Spectral sequences have generally been thought of as being complicated, esoteric constructions, due mainly to the way they are presented in the mathematical literature. This book is very unusual, in that it attempts to explain the need for spectral sequences and give insight into how they arise and in what contexts. Anyone who is curious about spectral sequences will find an exceptionally well-written book here. This goes especially for the physicist reader, who if involved in fields such as string theory or quantum field theory, is faced with a daunting task of learning both the physics and mathematics behind these theories, formidable as both of these are. Chapter one, entitled `An Informal Introduction', is one of the best introductions to spectral sequences in print, in both books and research papers. The intuition gained by the reading of this chapter is invaluable for the chapters that follow, since the author motivates the construction of spectral sequences exceedingly well, with many examples given.
The author introduces spectral sequences as a tool for computing the homology or cohomology (which he labels as H*) of a space or an algebraic invariant assigned to a space or algebraic object. In order to obtain a more tractable problem and to motivate the calculation of H* using spectral sequences, the author assumes at first that H* is `filtered', in particular that H* is a graded vector space. As a first approximation to H*, one uses the associated graded vector space to some filtration of H*, which is the "target" of the spectral sequence. The "two-index" property of spectral sequences in this case arises from the fact that the associated graded vector space to the filtered graded vector space is in fact `bigraded'. One of the indices is called the `complementary degree' while the other is called the `filtration degree.' More formally, the spectral sequence is a sequence of differential bigraded vector spaces, where each bigraded vector space in the sequence is equipped with a linear mapping that is also a differential. The goal is then to find the conditions under which the spectral sequence will `converge' to H*. In the introductory chapter, the author outlines various situations that allow one to compute with a spectral sequence. Some familiar constructions appear, such as the Gysin sequence, known from homological algebra and differential geometry, and the exterior algebra, also from differential geometry.
With the motivation for spectral sequences established in the introduction, the author proceeds to more formal constructions in the next chapter. Spectral sequences arise as a collection of differential bigraded R-modules between which are defined differentials. The author shows in detail how to build spectral sequences using a filtered differential module and using an exact couple. As per the historical development, he also constructs spectral sequences of algebras using tensor products of differential graded modules. After these constructions are made, the author turns his attention to how well the spectral sequence can approximate its target. This entails, as expected, a rigorous notion of limits. The author in fact defines limits and colimits of modules and the notion of a morphism between spectral sequences. For filtered differential graded modules, he shows how conditions on the filtration will ensure the associated spectral sequence converges uniquely to its target. For exact couples, the convergence can be shown but certain properties such as the Hausdorff property for the filtration must be satisfied.
The book covers four main spectral sequences that arise in algebraic topology: the Leray-Serre, Eilenberg-Moore, Adams, and Bockstein spectral sequences. The Leray-Serre spectral sequence arises when studying the homology (and cohomology) of fibrations with path-connected base spaces and connected fibers. The Leray-Serre spectral sequence allows one to compute the cohomology of the total space from knowledge of the cohomology of the base space and the fiber. The author discusses applications in the computation of cohomology of Lie groups. This is accomplished by constructing the fibration resulting from taking quotients by subgroups. Rigorous proofs of all the constructions are given for the interested reader, including a full proof of the theorem that the fourth homotopy group of the two-sphere is the integers modulo two, and the connections with characteristic classes and the Steenrod algebra.
The Eilenberg-Moore spectral sequence also arises in the study of fibrations, when the cohomology of the base space and the cohomology of the total space are known and one wants to compute the cohomology of the fiber. The author studies this case and the dual case of the Eilenberg-Moore spectral sequence for homology. Heavy use is made of differential homological algebra in this study. The reader can see with great clarity the role of torsion in the applications of the Eilenberg-Moore spectral sequences.
The Adams spectral sequence arises in the context of computing the homotopy groups of a nontrivial finite CW-complex. An approximation to the homotopy groups is given by the `stable homotopy groups', and Adams analysis of these groups and his proof that there are no elements of Hopf invariant one led him to construct the spectral sequence that bears his name. The author gives a detailed overview of this spectral sequence, its applications, and its connection with cobordism theory.
The Bockstein spectral sequence arose in the study of Lie groups, and the author gives the details of the construction of this spectral sequence and its application to H-spaces. Bockstein spectral sequences arise from exact couples, the first differential being the Bockstein homomorphism (in the case of homology). The Bockstein spectral sequence can also be constructed for the case of cohomology, wherein the Bockstein homomorphism becomes the stable cohomology operation in the Steenrod algebra. The resulting spectral sequence is in fact a spectral sequence of algebras with the stable cohomology operation being a derivation with respect to the cup product.
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Homology (Classics in Mathematics)
Saunders MacLane
Manufacturer: Springer
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ASIN: 3540586628 |
Book Description
This classic and much-cited book is a systematic introduction to homological algebra, starting with basic notions in abstract algebra and category theory and continuing with an up-to-date treatment of various advanced topics. Although the subject depends on the use of very general ideas, the book proceeds from the special to the general. The main ideas are introduced gradually with many examples illustrating why they are needed and what they can do. In conclusion the book treats additive functors in an abelian category relative to a proper class of exact sequences subsuming earlier results. The author has added many historical notes and also exercises which are designed both to give elementary practice in the concepts presented and to formulate additional results not included in the text.
Book Description
This book is designed to be an introduction to some of the basic ideas in the field of algebraic topology. In particular, it is devoted to the foundations and applications of homology theory. The only prerequisite for the student is a basic knowledge of abelian groups and point set topology. The essentials of singular homology are given in the first chapter, along with some of the most important applications. In this way the student can quickly see the importance of the material. The successive topics include attaching spaces, finite CW complexes, the Eilenberg-Steenrod axioms, cohomology products, manifolds, Poincaré duality, and fixed point theory. Throughout the book the approach is as illustrative as possible, with numerous examples and diagrams. Extremes of generality are sacrificed when they are likely to obscure the essential concepts involved. The book is intended to be easily read by students as a textbook for a course or as a source for individual study. The second edition has been substantially revised. It includes a new chapter on covering spaces in addition to illuminating new exercises.
Customer Reviews:
Has the good and bad.......2005-12-30
This is a terrific book on homology theory, covering all the standard topics, plus some nice topics that are hard to find in other introductory books. The motivation for theory is presented in both algebraic/categorical and geometric flavors. The structure of the book is mostly solid, getting straight to the point with singular homology instead of wasting time with simplicial homology and its results (a rarity with algebraic topology books). My only complaints are that the book is riddled with typos and chapter 5 (on products in homology and cohomology) is quite messy.
Masterful.......2000-10-01
This introduction to singular homology combines a strong historical sense with an easy mastery of modern methods. The massive contributions of Poincare and Brouwer are credited, and their geometrical motivations are clear. At the same time the book neither minimizes nor apologizes for modern algebraic machinery, but treats categories and acyclic models and more as natural means to simplify the subject. The book goes through Poincare duality and a good account of the Lefschetz fixed point theorems. It is at once very visual and algebraically slick. The only problem with this approach is that the author seems a bit uncomfortable descending into the nuts and bolts of the longer proofs of two key results (the acyclic model theorem, and the duality theorem). He handles the details unevenly and makes some actual mis-statements. Here the reader needs the experience and confidence to make some corections.
Average customer rating:
- A below average text book
- Sometimes annoying but instructive
- A bad book for 7 th graders like me
- Opinión general
- Good book for applications of fractal geometry, but....
|
Fractals Everywhere
Michael F. Barnsley
Manufacturer: Morgan Kaufmann
ProductGroup: Book
Binding: Paperback
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SuperFractals
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The Fractal Geometry of Nature
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Introducing Fractal Geometry, 3rd Edition (Introducing)
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Chaos and Fractals: New Frontiers of Science
ASIN: 0120790696 |
Book Description
This volume is the second edition of the highly successful Fractals Everywhere. The Focus of this text is how fractal geometry can be used to model real objects in the physical world.
This edition of Fractals Everywhere is the most up-to-date fractal textbook available today.
Fractals Everywhere may be supplemented by Michael F. Barnsley's Desktop Fractal Design System (version 2.0) with IBM for Macintosh software. The Desktop Fractal Design System 2.0 is a tool for designing Iterated Function Systems codes and fractal images, and makes an excellent supplement to a course on fractal geometry
* A new chapter on recurrent iterated function systems, including vector recurrent iterated function systems.
* Problems and tools emphasizing fractal applciations.
* An all-new answer key to problems in the text, with solutions and hints.
Customer Reviews:
A below average text book.......2006-11-13
Barnsley's book has a shortcoming common to many math text books -- it's poorly written. Barnsely's writing style is superfluous and rambling. What I learned from this book was in spite of Barnsley's writing, not because of it.
Furthermore, the book's illustrations are substandard. There are over five different fonts used in illustrations (including hand written text). This leads to confusion when you're unsure if the text in Barnsley's illustrations is referring to Greek letters or the conventional alphabet. Another shortcoming is that Barnsley intermingles end of chapter exercises with new concepts. You may have problems 1.1 to 1.5 reviewing what you've already learned, and then problem 1.6 introduces completely new material. This is a problem throughout the book, as important concepts are introduced in exercises or otherwise illogical locations.
On the positive side, solutions to most exercises are presented at the end of the book. Overall the book was useful, but learning the material was unnecessarily difficult due to the book's shortcomings.
Sometimes annoying but instructive.......2002-08-27
Although instructive, this book is sometimes annoying to read. The author seems to be playing his cards very close to the vest and not telling us everything.
For instance, there is little or no instruction on how to implement the IFS attractors presented as a panacea for data compression. This seems to be proprietary to his company. It also seems that hands-on manipulation is crucial to the images' production, contrary to the author's claims.
If you can understand the mathematics you may find the book useful, as I did when writing my book Fractals in MUsic.
A bad book for 7 th graders like me.......2001-11-28
this is a bad and very confusing book for a young student in, say... 7th grade, like me. The language is incomprehensible and there are no visual aids.
Opinión general.......2001-11-19
HUmmm!! parece interesante este librito. Pero la verdad busco uno donde encuentre aplicaciones a la ingeniería.
Estos libros de teoría suelen ponerse aburridos al no tener sufuciente información sobre aplicaciones.
De todos modos apenas lo tenga en las manos y lo mire doy una opinión más seria de este.
Good book for applications of fractal geometry, but...........2001-06-23
This is a good book on applications of fractals; the chapters on modeling natural objects with iterated function systems (IFS) and fractal interpolation are especially useful. Many standard topics are included, for example, fractal dimension, Julia and Mandelbrot sets, chaos and the shift dynamical system. Some of the illustrations are captivating.
However, the book is not well organized, and the writing is extremely wordy to the point of being irritating. Some paragraphs read as if they belonged to a "Dummies" handbook. Also, I have to agree with one reviewer that the treatment of fractal dimension is poor. For one thing, it does not fully develop the intuition behind the concept-- much less the math. This same remark holds for the chapter on chaotic dynamics.
In summary, the book is fine for applications, but supplement your reading with a more substantial text.
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