Product 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
Contents: Introduction. - Fundamental Concepts. -Topological Vector Spaces.- The Quotient Topology. -Completion of Metric Spaces. - Homotopy. - The TwoCountability Axioms. - CW-Complexes. - Construction ofContinuous Functions on Topological Spaces. - CoveringSpaces. - The Theorem ofTychonoff. - Set Theory (by T.Br|cker). - References. - Table of Symbols. -Index. ... Read more

Customer Reviews (5)

great as motivation but not a textbook
While I agree with the other reviewers here that Jaenich's "Topology" is very well written, goes to great lengths to explain the "hows and whys" of topology, and includes many, many figures (about 1 per page on average), it is probably more popular with people who already know topology than with beginning students, even though it is an introductory text intended for undergraduates. This is due to both a frequent lack of precision or formality in proofs and definitions coupled with a tendency to discuss much more advanced material with which a student at this level wouldn't be familiar. I believe that experienced mathematicians, who perhaps learned point-set topology from books such as those of Munkres, Kelley, or Bredon (or even an analysis book such as Royden), appreciate how this book focuses on motivating the concepts, explaining how the various objects are used elsewhere in mathematics - for that purpose this is one of the finest books I have seen. However, too much material is mentioned that is certainly over the heads of most students new to topology, such as the Pontrjagin-Thom construction, the spectrum of commutative Banach algebras, or Lie groups, often in a very cursory manner that would serve only to confuse beginners. Concepts are often used before they are defined, or are not defined precisely, which is liable to frustrate these students as well. Many topics are given such short attention it makes you wonder why the author even bothered - such as a page devoted to Frechet spaces followed by a section consisting of a single paragraph on locally convex topological vector spaces. Much of the material is not covered very deeply - only a definition and maybe a theorem, which half the time isn't even proved but just cited.

Certainly this book couldn't be used as a textbook for an undergraduate course - for the reasons mentioned above and also because not enough material is actually covered, as well as the obvious deficiency in that it lacks exercises for the reader. Most of the proofs until the last chapters are of the 1- or 2-paragraph variety, with some pictures added, although as the book progresses the level becomes increasingly more sophisticated. The book also covers both point-set topology (topological spaces, compactness, connectedness, separation axioms, completeness, metric topology, TVS, quotient topology, countability, metrization, etc.) and elements of algebraic topology (homotopy, fundamental group, simplices, CW-complexes, covering spaces, but not really homology), but the presentation of the algebraic topology in particular is not liable to be helpful for the novice, except for the treatment of covering spaces, which is perhaps the highlight of the book. Half of the chapter on homotopy is actually concerned with categories and functors, probably not the best way to introduce the subject. In fact, here is direct quote from the index:
"We talk about homology (and a number of other objects beyond the realm of point-set topology) several times in this book, but the definition is not given."
That, in a nutshell, explains the difficulty with this book.

So why am I rating this 5 stars? For the wealth of examples (e.g., 4 sections on examples of quotient spaces) and explanations of how these concepts are used and why they are important. Just by looking at the contents one can see this, as there are sections titled:
"What is point-set topology about?," "What is algebraic topology?," "Homotopy - what for?," "The role of the countability axioms," "Why CW-complexes are more flexible," "Yes, but...?," "The role of covering spaces in mathematics," "What is it [Tychonoff's theorem] good for?"
The chapter on covering spaces, coming near the end of the book, is particularly good, with a proof of their classification given. This is definitely the most fleshed-out part; if only the rest of the book could go into this depth.

This book would make an excellent supplement to a more formal textbook such as Munkres, but is not a substitute for it. But I would still consider this as a must-read for all those students who plan on studying mathematics in graduate school.

A simple introduction to advanced mathematical concepts
This text gives the reason behind many advanced topological concepts and tantalizes the reader with it's varied applications.

Basic topological concepts of open, closed, continuous, product topology, connectedness,compactness and intro to separation axioms is presented in a logical concise and easy to understand way.

The author then delves into topological groups and vector spaces introducting Hilbert Banach and Frechet spaces ( albeit briefly ).

Quotient spaces,homotopy, complexes and urysohn and tietze lemma along with partitions of unity are tackled next.

I especially enjoyed the section on covering spaces with which it concludes.

Perhaps the single best accolade I can give the book is that it gives one inspiration and motivation to explore in greater detail mathematical objects discussed.

The text is useful to all students of mathematics and physics alike.

Full of motivations
This book is fun to read. In a weekly homework meeting for an Algebraic Geometry class, I complained to one grad student "Geometry textbooks should have many pictures", and he asked "Define 'many'?" I said "One on each page". Now this topology book is certainly close to that. (It has more than 180 illustrations.) Though its written style is a bit informal, 'handwaving' arguments can serve as outlines of rigorous proofs.

Since it does not have any problem sections, I can see why Munkres' book is more popular in college. It still gives some inspiring questions from time to time. Besides the basic pot-set topology, it also covers some algebraic topology and differentialtopology. The author does not hesitate to use examples from those advanced areas without formal definitions, and this was a bit annoying when I read it the first time. In this sense, the book is not really selfcontained. However, when finally a notion is formally defined, I can see it from various aspects in those examples. This really helps me understand topology better, and makes me want to explore them. After reading the existence thm of covering spaces in chapter 9, I realized that mathematics is really an art.

The index in the back of the book is in the format of short definitions, which can be used as a quick reference.

Students: BUY THIS BOOK!!!
It is not too often that a book about topology is written with the goal of actually explaining in detail what is going on behind the formalism. The author does a brilliant job of teaching the reader the essential concepts of point set topology, and the book is very fun to read. The reader will walk away with an appreciation of the idea that topology is just not abstract formalism, but has an underlying intuition that is rich in imagery. The author has a knack for allowing readers to "see into the future" of what kind of mathematics is waiting for them and how topology is indispensable in its study.

At the end of chapter three, which deals with the quotient topology, the author writes the following paragraph: "If is often said against intuitive, spatial argumentation that it is not really argumentation but just so much gesticulation - just 'handwaving'. Shall we then abandon all intuitive arguments? Certainly not. As long as it is backed by the gold standard of rigorous proofs, the paper money of gestures is an invaluable aid for quick communication and fast circulation of ideas. Long live handwaving!". This has to rank as one of the best paragraphs that has every appeared in a mathematics book, for it nicely summarizes the need for developing a feel for the concepts behind mathematics before moving on to the rigorous proofs. Physicists in particular, who must assimilate mathematics very quickly in order to apply it to real problems must have a pictorial, "playful" understanding of the mathematical constructions.

Thus the language that the author employs is informal, and a listing of the best discussions in the book would really entail a listing of every one in the book. There is not one part of the book that is not helpful or interesting, and the author delves into many different areas that involve the use of topology.

If you are a beginning student in mathematics, BUY AND STUDY THIS BOOK...BUY AND STUDY THIS BOOK. You will take away so much for the price paid.

Excellent
Excelent, clear, well-motivated introduction ... Read more

Product Description
This self-contained introduction to algebraic topology is suitable for a number of topology courses. It consists of about one quarter 'general topology' (without its usual pathologies) and three quarters 'algebraic topology' (centred around the fundamental group, a readily grasped topic which gives a good idea of what algebraic topology is). The book has emerged from courses given at the University of Newcastle-upon-Tyne to senior undergraduates and beginning postgraduates. It has been written at a level which will enable the reader to use it for self-study as well as a course book. The approach is leisurely and a geometric flavour is evident throughout. The many illustrations and over 350 exercises will prove invaluable as a teaching aid. This account will be welcomed by advanced students of pure mathematics at colleges and universities. ... 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

Product Description
Since it was first published in 1967, Simplicial Objects in Algebraic Topology has been the standard reference for the theory of simplicial sets and their relationship to the homotopy theory of topological spaces.J. Peter May gives a lucid account of the basic homotopy theory of simplicial sets (discrete analogs of topological spaces) which have played a central role in algebraic topology ever since their introduction in the late 1940s.

"Simplicial Objects in Algebraic Topology presents much of the elementary material of algebraic topology from the semi-simplicial viewpoint.It should prove very valuable to anyone wishing to learn semi-simplicial topology.[May] has included detailed proofs, and he has succeeded very well in the task of organizing a large body of previously scattered material."--Mathematical Review ... Read more

Customer Reviews (1)

Fast delivery
The book was in the same conditions as described when I bought it and it came in time. ... Read more

Product Description
In recent years, for solving problems of algebraic topology and, in particular, difficult problems of homotopy theory, algebraic structures more complicated than just a topological monoid, an algebra, a coalgebra, etc., have been used more and more often. A convenient language for describing various structures arising naturally on topological spaces and on their cohomology and homotopy groups is the language of operads and algebras over an operad. This language was proposed by J. P. May in the 1970s to describe the structures on various loop spaces. This book presents a detailed study of the concept of an operad in the categories of topological spaces and of chain complexes. The notions of an algebra and a coalgebra over an operad are introduced, and their properties are investigated. The algebraic structure of the singular chain complex of a topological space is explained, and it is shown how the problem of homotopy classification of topological spaces can be solved using this structure. For algebras and coalgebras over operads, standard constructions are defined, particularly the bar and cobar constructions. Operad methods are applied to computing the homology of iterated loop spaces, investigating the algebraic structure of generalized cohomology theories, describing cohomology of groups and algebras, computing differential in the Adams spectral sequence for the homotopy groups of the spheres, and some other problems. ... Read more

Product Description
Excellent text offers comprehensive coverage of elementary general topology as well as algebraic topology, specifically 2-manifolds, covering spaces and fundamental groups. The text is accessible to students at the advanced undergraduate or graduate level who are conversant with the basics of real analysis or advanced calculus. Problems, with selected solutions. Bibliography. 1975 edition.
... Read more

Customer Reviews (5)

Good book / Great supplement
I used this book as a supplement for math590(ind.study) on simplicial homology groups. This book is good because its cheap, concise, and rigorous (and characterizes all compact surfaces as well as proves the simplicial approximation theorem). There are hints and answers to the exercises which are numerous and vary in difficulty. I think this is the best all around book Dover offers on topology; Maunders is completely in a league of its own.

Who cares who writes the reviews
It is the author's ability to properly convey the ideas of the given subject that should be addressed not who the author chose to review it.Having read books by several of the "well-known' authors in the area of Topology, i.e. Munkres, Kelley and Bourbaki, I found this book quite informative, lively and lives up to the author's assertion that there is a definte need for books that are less dense(terse, pedantic) and which get right to the point, illustrating and presenting the material essential for an introductory exposure to Topology.The exercises are well chosen and extend the material presented in the text which is a complementary bonus since there appears an unfortunate trend in some texts to have seemingly irrelevant exercises at the end of each section.Also I found this to be good book for independent study and strongly recommend it to all highly motivated undergraduates.

Not Funny, Professor
While a good sense of humor is as necessary to reading (and hopefully to learning) topology as is mental acuity, self-serving Reviews by Authors are offensive and similar "contributions" by Authors' relatives (ifthat is in fact what we have here)are even more so given the price oftextbooks these days. Why doesn't the good professor submit the results ofa survey of his students (anonymous and conducted by someone other thanhimself, we would hope)who have used this book as a text? Better yet, wouldthe professor submit an open letter to a peer-reviewed journal solicitingsuch evaluations from colleagues at other schools which haveused his text?

Excellent Text--Dad you were right.
I find this to be an excellent treatment of concepts with have been made quite difficult in other texts.

Dad approves!
As the father of the author, I recommend this book heartily to topologists the world over.I originally purchasedmy copy at Blackwell's bookshop in Oxford, England. --Irving Kah ... Read more

Product Description
This set of notes, for graduate students who are specializing in algebraic topology, adopts a novel approach to the teaching of the subject. It begins with a survey of the most beneficial areas for study, with recommendations regarding the best written accounts of each topic. Because a number of the sources are rather inaccessible to students, the second part of the book comprises a collection of some of these classic expositions, from journals, lecture notes, theses and conference proceedings. They are connected by short explanatory passages written by Professor Adams, whose own contributions to this branch of mathematics are represented in the reprinted articles. ... Read more

Product Description
This book arose from courses taught by the authors, and isdesigned for both instructional and reference use during and after afirst course in algebraic topology. It is a handbook for users whowant to calculate, but whose main interests are in applications usingthe current literature, rather than in developing the theory. Typicalareas of applications are differential geometry and theoreticalphysics. We start gently, with numerous pictures to illustrate the fundamentalideas and constructions in homotopy theory that are needed in laterchapters. We show how to calculate homotopy groups, homology groupsand cohomology rings of most of the major theories, exact homotopysequences of fibrations, some important spectral sequences, and allthe obstructions that we can compute from these. Our approach is tomix illustrative examples with those proofs that actually developtransferable calculational aids. We give extensive appendices withnotes on background material, extensive tables of data, and a thoroughindex. Audience: Graduate students and professionals in mathematics andphysics. ... Read more

Product Description
This book is written as a textbook on algebraic topology. The first part covers the material for two introductory courses about homotopy and homology. The second part presents more advanced applications and concepts (duality, characteristic classes, homotopy groups of spheres, bordism). The author recommends starting an introductory course with homotopy theory. For this purpose, classical results are presented with new elementary proofs. Alternatively, one could start more traditionally with singular and axiomatic homology. Additional chapters are devoted to the geometry of manifolds, cell complexes and fibre bundles. A special feature is the rich supply of nearly 500 exercises and problems. Several sections include topics which have not appeared before in textbooks as well as simplified proofs for some important results. Prerequisites are standard point set topology (as recalled in the first chapter), elementary algebraic notions (modules, tensor product), and some terminology from category theory. The aim of the book is to introduce advanced undergraduate and graduate (master's) students to basic tools, concepts and results of algebraic topology. Sufficient background material from geometry and algebra is included. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. ... Read more

Product Description
This book presents a geometric introduction to the homology of topological spaces and the cohomology of smooth manifolds. The author introduces a new class of stratified spaces, so-called stratifolds. He derives basic concepts from differential topology such as Sard's theorem, partitions of unity and transversality. Based on this, homology groups are constructed in the framework of stratifolds and the homology axioms are proved. This implies that for nice spaces these homology groups agree with ordinary singular homology. Besides the standard computations of homology groups using the axioms, straightforward constructions of important homology classes are given. The author also defines stratifold cohomology groups following an idea of Quillen. Again, certain important cohomology classes occur very naturally in this description, for example, the characteristic classes which are constructed in the book and applied later on. One of the most fundamental results, Poincare duality, is almost a triviality in this approach. Some fundamental invariants, such as the Euler characteristic and the signature, are derived from (co)homology groups. These invariants play a significant role in some of the most spectacular results in differential topology. In particular, the author proves a special case of Hirzebruch's signature theorem and presents as a highlight Milnor's exotic 7-spheres. This book is based on courses the author taught in Mainz and Heidelberg. Readers should be familiar with the basic notions of point-set topology and differential topology. The book can be used for a combined introduction to differential and algebraic topology, as well as for a quick presentation of (co)homology in a course about differential geometry. ... Read more

Product Description

This book brings the most important aspects of modern topology within reach of a second-year undergraduate student. It successfully unites the most exciting aspects of modern topology with those that are most useful for research, leaving readers prepared and motivated for further study. Written from a thoroughly modern perspective, every topic is introduced with an explanation of why it is being studied, and a huge number of examples provide further motivation. The book is ideal for self-study and assumes only a familiarity with the notion of continuity and basic algebra.

Customer Reviews (4)

Best Intro to Topology
Topics are well motivated.
Theorems are proved in a rigorous yet intuitive style that one feels like it was an explanation rather than a dry proof typically found in the advanced math books.
Important key ideas are also sufficiently illustrated through examples and exercises.

If one finds it verbose, I'd recommendcroom--a bit more like the typical math books but accessible.

Surely not optimal
We used this book in an introductory topology class I took.Some of the exercises are poor (e.g. counting the number of topologies) and the exposition wasn't anything to go crazy about.After a while, I found myself reading Munkres exclusively; it's much more comprehensive.Maybe this book is well suited for folks looking to get a flavor of topology but nothing super concrete.

Best undergraduate topology book
I have never seen such a beatiful explanation on continuity and its relations to series and sets. Now I understand why, when mathematics is lousily explained,everything seemms to be so hard. I recommend strongly this book for someone for self study on topology. Hope the author can write on other topics of mathematics.

A pleasure to read
I have a major in math, many years ago. I have moved into economics, but miss the elegance of math, hence I decided to revisit some old topics, and started with topology. As a student we used lecture notes and no real textbook, so my choice now was this textbook. It is a pure pleasure to read. I wish we had used it as a text book when I studied.

The topics are well motivated. Crossley does a good job in explaining why we should care about these particular lemmas and theorems. The proofs are usually elegant. I find the estetic pleasures a good math book should provide. ... Read more

Customer Reviews (1)

Part 2, Singular Homology Theory is recommended.
This text is suitable for students of mathematics without prior knowledge of algebraic topology. The best thing with this is Part 2 which treats singular homology theory. However, you may want to resort to Maunder for aneffeective introductin to elelmentary homotopy theory, and to Dold for andintruduction to orientation and duality. ... Read more

Product 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

Product Description
This Introduction to Topology, which is a thoroughlyrevised, extensively rewritten, second edition of the work firstpublished in Russian in 1980, is a primary manual of topology. Itcontains the basic concepts and theorems of general topology andhomotopy theory, the classification of two-dimensional surfaces, anoutline of smooth manifold theory and mappings of smooth manifolds.Elements of Morse and homology theory, with their application to fixedpoints, are also included. Finally, the role of topology inmathematical analysis, geometry, mechanics and differential equationsis illustrated.
Introduction to Topology contains many attractive illustrationsdrawn by A. T. Frenko, which, while forming an integral part of thebook, also reflect the visual and philosophical aspects of moderntopology. Each chapter ends with a review of the recommendedliterature.
Audience: Researchers and graduate students whose work involvesthe application of topology, homotopy and homology theories.
... Read more

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"The author has attempted an ambitious and most commendable project. He assumes only a modest knowledge of algebraic topology on the part of the reader to start with, and he leads the reader systematically to the point at which he can begin to tackle problems in the current areas of research centered around generalized homology theories and their applications. ... The author has sought to make his treatment complete and he has succeeded. The book contains much material that has not previously appeared in this format. The writing is clean and clear and the exposition is well motivated. ... This book is, all in all, a very admirable work and a valuable addition to the literature ..."- S.Y. HUSSEINI IN MATHEMATICAL REVIEWS ... Read more

Customer Reviews (1)

This might take a while...
The earlier chapters are quite good; however, some ofthe advanced topics in this book are better approached (appreciated) after one has learned about them elsewhere,at a more leisurely pace.For instance, this isn't the best place to first read about characteristic classes and topological K theory (I would recommend, without much hesitation, the books by Atiyah and Milnor & Stasheff, instead).Much to my disappointment, the chapter on spectral sequences is quite convoluted. Parts of 'user's guide' by Mclearywould certainlycome in handy here (which sets the stage rather nicely for applications).

So it turns out that supplemental reading (exluding Whitehead's massive treatise) is necessary to achieve a better understanding of algebraic topology at the level of this book.The homotopical view therein will be matched (possibly superseded) by Aguilar's book (forthcoming, to which I am very much looking forward).

Good luck! ... Read more

Product Description
This volume discusses results about quadratic forms that give rise to interconnections among number theory, algebra, algebraic geometry, and topology. The author deals with various topics including Hilbert's 17th problem, the Tsen-Lang theory of quasi-algebraically closed fields, the level of topological spaces, and systems of quadratic forms over arbitrary fields. Whenever possible, proofs are short and elegant, and the author has made this book as self-contained as possible. This book brings together thirty years' worth of results certain to interest anyone whose research touches on quadratic forms. ... Read more

Product Description
Designed to be an introduction to some of the basic ideas in the field of algebraic topology. Devoted to the foundations and applications of homology theory. DLC: Homology theory. ... Read more

Customer Reviews (3)

(Co)Homology the Way it Should Be
This was the textbook for the first third of a year-long algebraic topology sequence at Oregon State in 1973-4. We were told by the prof that Vick was a student of Stong and that the book was essentially Stong's course written up with his blessing. It's hard to ask for a better pedigree than that, as Stong was a legend for his teaching (as well as his research).

Although there are some minor quibbles (noted in the 4-star reviews), I still haven't found a better treatment of the key results, nor a more direct path. The proof of Poincare duality in particular is that of Hans Samelson, another legend in the field for both teaching and research.

Checking the references, one finds that this was not the only such example where Vick sought out what was then regarded as the best proof available for beginners. It is also noteworthy that community consensus on which are best has not changed much, if any, since then.

The plethora of typos may be a "feature" of the reprint, since I don't recall that many in the original Acad. Press edition we used, and I still have.

As should be clear, this one is a real keeper.

For more modern/advanced study, continue with Switzer and Brayton Gray. By then the journals should be reasonably accessible.

Has the good and bad
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
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. ... Read more