Book Description
In this book, two seemingly unrelated fields -- algebraic topology and robust control -- are brought together.The book develops algebraic/differential topology from an application-oriented point of view.The book takes the reader on a path starting from a well-motivated robust stability problem, showing the relevance of the simplicial approximation theorem and how it can be efficiently implemented using computational geometry. The simplicial approximation theorem serves as a primer to more serious topological issues such as the obstruction to extending the Nyquist map, K-theory of robust stabilization, and eventually the differential topology of the Nyquist map, culminating in the explanation of the lack of continuity of the stability margin relative to rounding errors. The book is suitable for graduate students in engineering and/or applied mathematics, academic researchers and governmental laboratories. ... Read more

Book Description
At the present time, the average undergraduate mathematics major finds mathematics heavily compartmentalized. After Calculus, students take courses in analysis and algebra, and depending on their interest, they take courses in special topics. If the student is exposed to topology, it is usually straightforward point set topology; if the student is exposed to geometry, it is usually classical differential geometry.

These notes are an attempt to break up this compartmentalization, at least in topology-geometry. What the student has learned in algebra and advanced calculus are used to prove some fairly deep results relating geometry, topology and group theory. The material studied includes De Rhams's theorem, the Gauss-Bonnet theorem for surfaces, the functional relation of fundamental group to covering space, and surfaces of constant curvature as homogeneous spaces. ... Read more

Customer Reviews (2)

Nice overview of Point-Set, Algebraic, and Differential Topology
I believe that one of the authors, I. Singer is a Fields Medalist, so this is co-written by one of the masters in the subject.It is one of the few books I'm aware of that covers point-set, algebraic, and differential topology.However, this is not in any way an exhaustive text.It is very spare and to the point.This is a small book packed with information.No exercises are included.It is not particularly advance in any one area of topology, so keep in mind that this is just an overview of all the main areas.However, just because it is not particularly advanced, does not mean it is easy to read for someone completely unfamiliar with the material.I tried reading the early chapters as a first introduction to topology, and didn't know what was going on.Reading of it got better when I read it side-by-side with the books by Munkres and Armstrong.This book is pretty abstract and dense, so it will read slowly.A lot of the finer details in the subjects are found in other texts.

Concise and modern
First, a small carp at Amazon: The book has two authors and since John Thorpe is the junior and by far the less famous, I suspect that he actually did most of the writing. Please give him his due!

This is a very dense book. While this makes for rough sledding for the first timer, it's also an exciting introduction to modern topology and geometry and a good first step for those interested in such things in physics as gauge theories and superstrings. It's worth the effort.

Starting with the basics of set theory,the first couple chapters take the reader through point set topology. The next couple chapters introduce algebraic topology. The rest of the book is about the algebraic topology of differentiable manifolds and a very clean, modern introduction to the classical differential geometry of surfaces. The only caveat is, as Spivak says, "a weird proof of the de Rham theorem" in Chapter 6. I'm torn about this. The proof in Warner's "Foundations of Differentiable Manifold and Lie Groups" is much cleaner and better lends itself to other applications, but involves lots of machinery. The proof in Singer and Thorpe is a lot less elegant, using the lowest level tools possible. This makes the learning curve shorter and may make the theorem more clear, but may also obscure the big picture. Much of the important work in algebraic topology over the next 20 years and theoretical physics up to now is related to this result. Though much of this work was developed by Singer with his collaborater Michael Atiyah, their approach is closer to Warner's than to that in Singer and Thorpe.

For any particular topic in this book, you can find sources that you'll undoubtably find more digestible. This is the only book that brings them all together. It's an audacious effort. ... Read more

Customer Reviews (1)

Very well written introduction.
The book is a very well written introduction to algebraic topology. Itpresents the main basic results, provides geometric insights andcomputational tools. After reading it one really feels confident and onewhishes to go further : the book really is what the title says. It istherefore perfectly suited for a first reading. ... Read more

Book Description
This book is an introduction to the remarkable work of VaughanJones and Victor Vassiliev on knot and link invariants and itsrecent modifications and generalizations, including amathematical treatment of Jones-Witten invariants. It emphasizesthe geometric aspects of the theory and treats topics such asbraids, homeomorphisms of surfaces, surgery of 3-manifolds(Kirby calculus), and branched coverings. This attractivegeometric material, interesting in itself yet not previouslygathered in book form, constitutes the basis of the last twochapters, where the Jones-Witten invariants are constructed viathe rigorous skein algebra approach (mainly due to the SaintPetersburg school).

Unlike several recent monographs, where all of these invariantsare introduced by using the sophisticated abstract algebra ofquantum groups and representation theory, the mathematicalprerequisites are minimal in this book. Numerous figures andproblems make it suitable as a course text and for self-study. ... Read more

Book Description

Combinatorial algebraic topology is a fascinating and dynamic field at the crossroads of algebraic topology and discrete mathematics. This volume is the first comprehensive treatment of the subject in book form. The first part of the book constitutes a swift walk through the main tools of algebraic topology, including Stiefel-Whitney characteristic classes, which are needed for the later parts. Readers - graduate students and working mathematicians alike - will probably find particularly useful the second part, which contains an in-depth discussion of the major research techniques of combinatorial algebraic topology. Our presentation of standard topics is quite different from that of existing texts. In addition, several new themes, such as spectral sequences, are included. Although applications are sprinkled throughout the second part, they are principal focus of the third part, which is entirely devoted to developing the topological structure theory for graph homomorphisms. The main benefit for the reader will be the prospect of fairly quickly getting to the forefront of modern research in this active field.

Book Description

Customer Reviews (3)

readable
proofs are very detailed almost pedagogic. the author always points out the goal first then digs in. the only flaw is its visual presentation - rather condensed type set. i think Pontryagin was a truly educational teacher. good for self study.

A cheap classic by one of the masters
This book covers, as the title says, the foundations of combinatorial topology. What that refers to, essentially, is the topology of polyhedra and the machinery of simplicial homology groups. The first part of the bookestablishes the basic facts about these topological spaces and aboutabstract complexes as well. The second part shows the topologicalinvariance of simplicial homology. The third features applications of thematerial to fixed point theory.

Throughout the book there is also arecurring digression on dimension theory, which culminates in a proof ofthe very non-trivial fact that manifolds have the appropriate topologicaldimension.

The only formal pre-requisites for this book are basiclinear algebra and point-set topology, as covered in any advanced calculusor elementary real analysis course.

The presentation is very conciseand lucid throughout. I can imagine some people not liking such anextremely concise style, but the whole book is so logical that it worksvery well.

A great little book
This is a beautiful little book on a very pretty subject. Lev here develops very comprehensively the basic properties of polyhedra, as well as many of their basic applications. The first chapter is spent on basic topological properties of these spaces, with some applications todimension, very explicitly set aside from the main development. The secondchapter focuses on proving the topological invariance of the Betti groups,with applications of the techniques used-again carefully set aside-todimension theory and Brouwer's fixed point theorem. The third chapter,titled Continuous Mappings and Fixed points, has as a goal proving a simplefixed point theorem. Some simple constructions and concepts of algebraictopology, such as homotopy and the cylinder construction, are introducedalong the way. Unfortunately, not too long after the publication ofthis book, the author began to devote his energies to applied mathematicsand preventing Jews from polluting Russian mathematics-but don't let thatinfluence your possible purchase of this book. ... Read more

Product Description
This is the Russian translation of this text. ... Read more

Book Description
This book contains the proceedings of the Real AlgebraicGeometry-Topology Conference, held at Michigan State Universityin December 1993. Presented here are recent results anddiscussions of new ideas pertaining to such topics as resolutiontheorems, algebraic structures, topology of nonsingular realalgebraic sets, and the distribution of real algebraic sets inprojective space. ... Read more

Book Description
In this volume the authors seek to illustrate how methods of differential geometry find application in the study of the topology of differential manifolds. Prerequisites are few since the authors take pains to set out the theory of differential forms and the algebra required. The reader is introduced to De Rham cohomology, and explicit and detailed calculations are present as examples. Topics covered include Mayer-Vietoris exact sequences, relative cohomology, Pioncare duality and Lefschetz's theorem. This book will be suitable for graduate students taking courses in algebraic topology and in differential topology. Mathematicians studying relativity and mathematical physics will find this an invaluable introduction to the techniques of differential geometry. ... Read more

Book Description
This book is both an introduction to K-theory and a text in algebra.These two roles are entirely compatible.On the one hand, nothing more than the basic algebra of groups, rings, and modules is needed to explain the clasical algebraic K-theory.On the other hand, K-theory is a natural organizing principle for the standard topics of a second course in algebra, and these topics are presented carefully here.The reader will not only learn algebraic K-theory, but also Dedekind domains, class groups, semisimple rings, character theory, quadratic forms, tensor products, localization, completion, tensor algebras, symmetric algebras, exterior algebras, central simple algebras, and Brauer groups. The presentation is self-contained, with all the necessary background and proofs, and is divided into short sections with exercises to reinforce the ideas and suggest further lines of inquiry. The prerequisites are minimal: just a first semester of algebra (including Galois theory and modules over a principal ideal domain).No experience with homological algebra, analysis, geometry, number theory, or topology is assumed.The author has successfuly used this text to teach algebra to first year graduate students. Selected topics can be used to construct a variety of one-semester courses; coverage of the entire text requires a full year. ... Read more

Customer Reviews (1)

Didactic perfection
Mathematics, particularly research mathematics, or mathematics that is close to the frontiers of research would be considerably easier to learn if mathematicians would both explain what they are going to do and explain what historical context motivates the problems or issues they are interested in. This would grant the needed insights and "intuitions" into the subject, which are absolutely necessary if one is to understand a particular mathematical topic in depth. In their papers, monographs, and textbooks, mathematicians could perhaps include at various places in the text some `fundamental insights' into the concepts that are being discussed. In addition, they could describe in detail what their goals are and what strategies they are going to use to solve the problems as they develop.

Unfortunately not many works of mathematics are written in this manner, and so those who wish to learn a given topic must frequently undertake time-consuming literature searches and solve myriads of exercises and problem sets in order to gain the needed insight. This takes large blocks of time, and poses an even greater challenge for those, such as physicists, who not must not only master the physics but also the mathematical formalism used to frame physical theories. Physicists would welcome, and even in many cases grab at straws to learn mathematics in a way that they need to in order to push forward the frontiers of their own subject.

This book though is very different, and is one of the best works of mathematics put in print in the last three decades. The author has given the reader a work that is not only mathematically rigorous but also fits the learning framework discussed above. There is no doubt that K-theory is a highly esoteric subject, but it can be learned much more easily by the study of this book. Within its covers there are myriads of fundamental insights that the author shares with the reader that make the learning of K-theory much more palatable and actually downright fun. It should not be thought however that the reader does not have to exercise a fair amount of cognition when wrestling with the intricacies of K-theory. This is true for K-theory as with other branches of mathematics, but those hungry for a true understanding of K-theory will deeply appreciate the author's efforts in this book.

The goal of K-theory is to generalize linear algebra, the latter of which deals with linear transformations on vector spaces over fields. K-theory tries to find out, and make rigorous, what constructions in linear algebra carry over when the field is replaced by a ring R and the vector space is replaced by a module over this ring. The first issue that must be dealt with is that of the concept of dimension, which for the case of a vector space is well defined (it is the cardinality of its basis). But an R-module does not necessarily have a basis. So the strategy deployed by the author is find the R-modules that do have a dimension. To find out what it means for an R-module to have a basis, the author constructs R-modules that are generated by elements that are not "linearly related." These are called `free' R-modules and the generating set is called an `R-basis.' The author then characterizes free modules that have a finite basis. That this is a non-trivial exercise is proven by the fact that every nontrivial finite Abelian group is a Z-module, where Z is the ring of integers. In addition, every set is a basis of a free R-module so one must find `presentations' of R-modules. These allow the construction of R-modules satisfying certain needed properties. And then, as expected if one is to extend linear algebra, the author constructs `matrices' of R-linear maps between finitely generated free R-modules. However, a free R-module can have a basis with unequal cardinality, and therefore the author finds those rings R whose free modules have unique dimension and those over which every module is free. The needed property is called an `invariant basis number' and it turns out that most rings have an invariant basis number. But some finitely generated R-modules the author points out are too "small" to be free, and so he finds the appropriate generalization of free modules. These are the famous `projective modules' and are the objects on which algebraic K-theory is based.

The designation of "projective" refers to the familiar notion of a projection in ordinary vector space theory, i.e. a linear, idempotent operator. The author describes projective modules as being the `direct summands' of free R-modules, and so to obtain the needed generalization of dimension he constructs Abelian monoids of R-modules under the direct sum operation. This involves finding a universal construction of an Abelian group from a semigroup and this leads to the famous Grothendieck group K0(R) of finitely generated projective R-modules. The finitely generated projective R-modules are `stably isomorphic' if they become equal in K0(R); they are `stably equivalent' if they become congruent in K0(R) modulo finitely generated free modules. The strategy then becomes that of adding R to two finitely generated R-modules to make them stably isomorphic (or stably equivalent). In addition, one must find out to what extent K0(R) determines whether a finitely generated projective R-module is free. This brings up the notion of an R-module being `stably free', and the author finds those stably free R-modules which are free. This involves the notion of the `matrix completion' of a ring and of "shortening" unimodular rows. The author also studies the connection of K0 with number theory, eventually showing that the projective class group is isomorphic to the ideal class group when R is a Dedekind domain.

K0(R) is an element of a sequence of Abelian groups associated to each ring R. To find K1(R), the author finds an analog of the row operations in ordinary linear algebra. The elements of K1(R), the `Bass-Whitehead group' are row-equivalence classes of invertible matrices. A group homomorphism from K1(R) to a group G is the analog of the determinant in ordinary linear algebra, and is often called the `Whitehead-Bass' determinant. K1(R) can be thought of as the "abelianization" of the general linear group GL(R). The elements of K2(R) consist of the relations among the generators of the group of row operations on a matrix. The "standard" relations among these operations give the `Steinberg group', and K2(R) is the center of this group. ... Read more

Book Description
The classification of algebraic surfaces is an intricate and fascinating branch of mathematics, developed over more than a century and still an active area of research today. In this book, Professor Beauville gives a lucid and concise account of the subject, expressed simply in the language of modern topology and sheaf theory, and accessible to any budding geometer. A chapter on preliminary material ensures that this volume is self-contained while the exercises succeed both in giving the flavor of the classical subject, and in equipping the reader with the techniques needed for research. The book is aimed at graduate students in geometry and topology. ... Read more

Book Description
J. Frank Adams had a profound influence on algebraic topology, and his works continue to shape its development. The International Symposium on Algebraic Topology held in Manchester during July 1990 was dedicated to his memory, and virtually all of the world's leading experts took part. This two-volume work constitutes the proceedings of the symposium. The articles contained here range from overviews to reports of work still in progress, as well as a survey and complete bibliography of Adams' own work. These proceedings form an important compendium of current research in algebraic topology, and one that demonstrates the depth of Adams' many contributions to the subject. Here in the first volume the theme is mainly unstable homotopy theory, homological and categorical algebra. The second volume is oriented toward stable homotopy theory, the Steenrod algebra and the Adams spectral sequence. ... Read more

Book Description

The first edition of this book presented the theory of linear algebraic groups over an algebraically closed field. The second edition, thoroughly revised and expanded, extends the theory over arbitrary fields, which are not necessarily algebraically closed. It thus represents a higher aim.

Book Description
Algebraic and Differential Topology presents in a clear, concise, and detailed manner the fundamentals of homology theory. It first defines the concept of a complex and its Betti groups, then discusses the topolgoical invariance of a Betti group. The book next presents various applications of homology theory, such as mapping of polyhedrons onto other polyhedrons as well as onto themselves. The third volume in L.S. Pontryagin's Selected Works, this book provides as many insights into algebraic topology for today's mathematician as it did when the author was making his initial endeavors into this field. ... Read more

Product Description
Modern topology uses very diverse methods. This book is devoted largely to methods of combinatorial topology, which reduce the study of topological spaces to investigations of their partitions into elementary sets, and to methods of differential topology, which deal with smooth manifolds and smooth maps. Many topological problems can be solved by using either of these two kinds of methods, combinatorial or differential. In such cases, both approaches are discussed.One of the main goals of this book is to advance as far as possible in the study of the properties of topological spaces (especially manifolds) without employing complicated techniques. This distinguishes it from the majority of other books on topology.The book contains many problems; almost all of them are supplied with hints or complete solutions. ... Read more