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1. A Concise Course in Algebraic Topology (Chicago Lectures in Mathematics) by J. P. May | |
Paperback: 254
Pages
(1999-09-01)
list price: US$24.00 -- used & new: US$20.52 (price subject to change: see help) Asin: 0226511839 Average Customer Review: Canada | United Kingdom | Germany | France | Japan | |
Editorial Review Product Description Algebraic topology is a basic part of modern mathematics, and some knowledge of this area is indispensable for any advanced work relating to geometry, including topology itself, differential geometry, algebraic geometry, and Lie groups. This book provides a detailed treatment of algebraic topology both for teachers of the subject and for advanced graduate students in mathematics either specializing in this area or continuing on to other fields. J. Peter May's approach reflects the enormous internal developments within algebraic topology over the past several decades, most of which are largely unknown to mathematicians in other fields. But he also retains the classical presentations of various topics where appropriate. Most chapters end with problems that further explore and refine the concepts presented. The final four chapters provide sketches of substantial areas of algebraic topology that are normally omitted from introductory texts, and the book concludes with a list of suggested readings for those interested in delving further into the field. Customer Reviews (9)
If U want generalization out to infinity, this is it for you, in algebraic topology basics.
The Title Says it All
The opposite of Hatcher
Lucid and elegant, but not for beginners
A Unique and Necessary Book However, as Willard points out, mathematics is learned by successive approximation to the truth. As you becomes more mathematically sophisticated, you should relearn algebraic topology to understand it the way that working mathematicians do. Peter May's book is the only text that I know of that concisely presents the core concepts algebraic topology from a sophisticated abstract point of view. To make it even better, it is beautifully written and the pedagogy is excellent, as Peter May has been teaching and refining this course for decades. Every line has obviously been thought about carefully for correctness and clarity. As an example, ones first exposure to singular homology should be concrete approach using singular chains, but this ultimately doesn't explain why many of the artificial-looking definitions of singular homology are the natural choices. In addition, this decidedly old-fashioned approach is hard to generalize to other combinatorial constructions. Here is how the book does it: First, deduce the cellular homology of CW-complexes as an immediate consequence of the Eilenberg-Steenrod axioms. Considering how one can extend this to general topological spaces suggests that one approximate the space by a CW-complex. Realization of the total singular complex of the space as a CW-complex is a functorial CW-approximation of the space. As the total singular complex induces an equivalence of (weak) homotopy categories and homology is homotopy-invariant, it is natural to define the singular homology of the original space to be the homology of the total singular complex. Although sophisticated, this is a deeply instructive approach, because it shows that the natural combinatorial approximation to a space is its total singular complex in the category of simplicial sets, which lets you transport of combinatorial invariants such as homology of chain complexes. This approach is essential to modern homotopy theory. ... Read more |
2. Algebraic Topology by Allen Hatcher | |
Paperback: 550
Pages
(2001-11-15)
list price: US$37.99 -- used & new: US$30.01 (price subject to change: see help) Asin: 0521795400 Average Customer Review: Canada | United Kingdom | Germany | France | Japan | |
Editorial Review Product Description Customer Reviews (19)
Terrible textbook
More Hand-Waving Than an Orchestral Conductor
Really bad as a "readable" texbookbut good reference
amazing book, but caveat emptor
excellent modern introduction |
3. Algebraic Topology (Volume 0) by Edwin H. Spanier | |
Paperback: 548
Pages
(1994-12-06)
list price: US$79.95 -- used & new: US$40.00 (price subject to change: see help) Asin: 0387944265 Average Customer Review: Canada | United Kingdom | Germany | France | Japan | |
Editorial Review Product Description Customer Reviews (6)
Pioneering text
For reference ONLY After a brief introduction to set theory, general topology, and algebra, homotopy and the fundamental group are covered in Chapter 1. Categories and functors are defined, and some examples are given, but the reader will have to consult the literature for an in-depth discussion. Homotopy is introduced as an equivalence class of maps between topological pairs. Fixing a base point allows the author to define H-spaces, but he does not motivate the real need for using pointed spaces, namely as a way of obtaining the composition law for the loops in the fundamental group. By suitable use of the reduced join, reduced product, and reduced suspension, the author shows how to obtain H-groups and H co-groups. The fundamental group is defined in the last section of the chapter, and the author does clarify the non-uniqueness of the fundamental group based at different points of a path-connected space. Covering spaces and fibrations are discussed in the next chapter. The author does a fairly good job of discussing these, and does a very good job of motivating the definition of a fiber bundle as a generalized covering space where the "fiber" is not discrete. The fundamental group is used to classify covering spaces. In chapter 3 the author gets down to the task of computing the fundamental group of a space using polyhedra. Although this subject is intensely geometrical. only six diagrams are included in the discussion. Homology is introduced via a categorical approach in the next chapter. Singular homology on the category of topological pairs and simplicial homology on the category of simplicial pairs. The author begins the chapter with a nice intuitive discussion, but then quickly runs off to an extremely abstract definition-theorem-proof treatment of homology theory. The discussion reads like one straight out of a book on homological algebra. This approach is even more apparent in the next chapter, where homology theory is extended to general coefficient groups. The Steenrod squaring operations, which have a beautiful geometric interpretation, are instead treated in this chapter as cohomology operations. The logic used is impeccable but the real understanding gained is severely lacking. General cohomology theory is treated in the next chapter with the duality between homology and cohomology investigated via the slant product. Characteristic classes, so important in applications, are discussed using algebraic constructions via the cup product and Steenrod squares. Characteristic classes do have a nice geometric interpretation, but this is totally masked in the discussion here. The higher homotopy groups and CW complexesare discussed in Chapter 7, but again, the functorial approach used here totally obscures the underlying geometrical constructions. Obstruction theory is the subject of Chapter8, with Eilenberg-Maclane spaces leading off the discussion. The author does give some motivation in the first few paragraphs on how obstructions arise as an impediment to a lifting of a map, but an explicit, concrete example is what is needed here. The last chapter covers spectral sequences as applied to homotopy groups of spheres. More homological algebra again, and the same material could be obtained (and in more detail) in a book on that subject.
Definitely not for beginners
Excellent reference, poor textbook That said, if you already know the subject Spanier'sbook is an excellent reference. Even here, though, you'll need to providesome details toward the ends of the later chapters. Each chapter starts outrelatively easily and works up to a crescendo, the treatment becomingterser and more advanced. I give it four stars (5 for mathematicalquality, 3 for usefulness as a text). The first three chapters deal withcovering spaces and fibrations; the middle three with (co)homology andduality; the last three with general homotopy theory, obstruction theory,and spectral sequences. Some of Serre's classical results on finitenesstheorems for homotopy groups are presented.
Excellent reference, poor textbook That said, if you already know the subject Spanier'sbook is an excellent reference.Even here, though, you'll need to providesome details toward the ends of the later chapters.Each chapter startsout relatively easily and works up to a crescendo, the treatment becomingterser and more advanced. I give it four stars (5 for mathematicalquality, 3 for usefulness as a text).The first three chapters deal withcovering spaces and fibrations; the middle three with (co)homology andduality; the last three with general homotopy theory, obstruction theory,and spectral sequences.Some of Serre's classical results on finitenesstheorems for homotopy groups are presented. ... Read more |
4. An Introduction to Algebraic Topology (Dover Books on Mathematics) by Andrew H. Wallace | |
Paperback: 208
Pages
(2007-02-27)
list price: US$12.95 -- used & new: US$8.30 (price subject to change: see help) Asin: 0486457869 Canada | United Kingdom | Germany | France | Japan | |
Editorial Review Product Description This self-contained treatment assumes only some knowledge of real numbers and real analysis. The first three chapters focus on the basics of point-set topology, after which the text proceeds to homology groups and continuous mapping, barycentric subdivision, and simplicial complexes. Exercises form an integral part of the text. 1961 edition. |
5. Algebraic Topology by C. R. F. Maunder | |
Paperback: 384
Pages
(1996-06-14)
list price: US$16.95 -- used & new: US$10.00 (price subject to change: see help) Asin: 0486691314 Average Customer Review: Canada | United Kingdom | Germany | France | Japan | |
Editorial Review Product Description Customer Reviews (3)
Solid
Shouldn't be your first text in algebraic topology. 1. Needs more pictures, especially for the simplicialhomology Chapter. 2. CW complexes should be covered before duality and not after. 3. Needs more examples and exercises. Overall, the book is very good, if you have already someexperience in Algebraic Topology. I found that the Croom'sbook "Basic concepts of Algebraic Topology" is an excellent first textbook. Too bad it is out of print, since it is very popular, every time I get it from the library, someone else recalls it. The combination of these two books probablyis the right thing to have: Maunder's book picks up whereCroom has left you.
Not bad. |
6. A History of Algebraic and Differential Topology, 1900 - 1960 (Modern Birkhäuser Classics) by Jean Dieudonné | |
Paperback: 648
Pages
(2009-06-09)
list price: US$69.95 -- used & new: US$55.67 (price subject to change: see help) Asin: 0817649069 Average Customer Review: Canada | United Kingdom | Germany | France | Japan | |
Editorial Review Product Description This book is a well-informed and detailed analysis of the problems and development of algebraic topology, from Poincaré and Brouwer to Serre, Adams, and Thom. The author has examined each significant paper along this route and describes the steps and strategy of its proofs and its relation to other work. Previously, the history of the many technical developments of 20th-century mathematics had seemed to present insuperable obstacles to scholarship. This book demonstrates in the case of topology how these obstacles can be overcome, with enlightening results.... Within its chosen boundaries the coverage of this book is superb. Read it! —MathSciNet Customer Reviews (1)
More than a mere "history". |
7. Algebraic Topology by William Fulton | |
Paperback: 430
Pages
(1995-07-27)
list price: US$49.95 -- used & new: US$32.00 (price subject to change: see help) Asin: 0387943277 Average Customer Review: Canada | United Kingdom | Germany | France | Japan | |
Editorial Review Product Description The first part of the book emphasizes relations with calculus and uses these ideas to prove the Jordan curve theorem. The study of fundamental groups and covering spaces emphasizes group actions. A final section gives a taste of the generalization to higher dimensions. Customer Reviews (3)
A book of ideas As a warm-up in Part 1, the author gives an overview of calculus in the plane, with the intent of eventually defining the local degree of a mapping from an open set in the plane to another. This is done in the second part of the book, where winding numbers are defined, and the important concept of homotopy is introduced. These concepts are shown to give the fundamental theorem of algebra and invariance of dimension for open sets in the plane. The delightful Ham-Sandwich theorem is discussed along with a proof of the Lusternik-Schnirelman-Borsuk theorem. I would like to see a constructive proof of this theorem, but I do not know of one. Part 3 is the tour de force of algebraic topology, for it covers the concepts of cohomology and homology. The author pursues a non-traditional approach to these ideas, since he introduces cohomology first, via the De Rham cohomology groups, and these are used to proved the Jordan curve theorem. Homology is then effectively introduced via chains, which is a much better approach than to hit the reader with a HOM functor.Part 4 discusses vector fields and the discussion reads more like a textbook in differential topology with the emphasis on critical points, Hessians, and vector fields on spheres. This leads naturally to a proof of the Euler characteristic. The Mayer-Vietoris theory follows in Part 5, for homology first and then for cohomology. The fundamental group finally makes its appearance in Part 6 and 7, and related to the first homology group and covering spaces. The author motivates nicely the Van Kampen theorem. A most interesting discussion is in part 8, which introduces Cech cohomology. The author's treatment is the best I have seen in the literature at this level. This is followed by an elementary overview of orientation using Cech cocycles. All of the constructions done so far in the plane are generalized to surfaces in Part 9. Compact oriented surfaces are classified and the second de Rham cohomology is defined, which allows the proof of the full Mayer-Vietoris theorem. The most important part of the book is Part 10, which deals with Riemann surfaces. The author's treatment here is more advanced than the rest of the book, but it is still a very readable discussion. Algebraic curves are introduced as well as a short discussion of elliptic and hyperelliptic curves. The level of abstraction increases greatly in the last part of the book, where the results are extended to higher dimensions. Homological algebra and its ubiquitous diagram chasing are finally brought in, but the treatment is still at a very understandable level. For examples of the author's pedagogical ability, I recommend his book Toric Varieties, and his masterpiece Intersection Theory.
This is one of the great algebraic topology books!
Probably better as a 2nd (or 3rd) course rather than 1st |
8. Algebraic Topology: An Introduction (Graduate Texts in Mathematics) (v. 56) by William S. Massey | |
Hardcover: 292
Pages
(1977-11-02)
list price: US$84.95 -- used & new: US$62.51 (price subject to change: see help) Asin: 0387902716 Average Customer Review: Canada | United Kingdom | Germany | France | Japan | |
Editorial Review Product Description Customer Reviews (1)
one of the best books on algebraic topology |
9. Algebraic Topology (Colloquium Pbns. Series, Vol 27) by Solomon Lefschetz, Solomon Lefschetz | |
Paperback: 389
Pages
(1980)
list price: US$48.00 -- used & new: US$47.36 (price subject to change: see help) Asin: 0821810278 Canada | United Kingdom | Germany | France | Japan | |
10. Combinatorial Algebraic Topology (Algorithms and Computation in Mathematics) by Dimitry Kozlov | |
Paperback: 390
Pages
(2008-01-30)
list price: US$59.95 -- used & new: US$39.37 (price subject to change: see help) Asin: 3540730516 Average Customer Review: Canada | United Kingdom | Germany | France | Japan | |
Editorial Review Product 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. Customer Reviews (1)
Help, where are the editors? |
11. Lectures on Algebraic Topology (Classics in Mathematics) by Albrecht Dold, A. Dold | |
Paperback: 400
Pages
(1995-02-15)
list price: US$59.95 -- used & new: US$44.00 (price subject to change: see help) Asin: 3540586601 Average Customer Review: Canada | United Kingdom | Germany | France | Japan | |
Editorial Review Product Description Customer Reviews (2)
Singular homology, products and manifolds
Elgant treatment of homology theory. |
12. Elements Of Algebraic Topology by James R. Munkres | |
Paperback: 464
Pages
(1996-01-01)
list price: US$76.00 -- used & new: US$68.40 (price subject to change: see help) Asin: 0201627280 Average Customer Review: Canada | United Kingdom | Germany | France | Japan | |
Customer Reviews (4)
Not bad..
A little incomplete However I think it is a little incomplete because of several reasons. (1)It pays no attention to one basic concept ofalgebraic topology: the fundamental group. (2) It doesn't cover ^Cechhomology, important in other areas, like dimension theory forexample. (3) It doesn't stress the most important feature of algebraictopology: its connection to other areas of mathematics (analysis,differential geometry, etc.). (4) Its list of references is too short,and lacks almost completely HISTORICAL references which are alwaysimportant to become an expert in any field. Conclusion: a good referenceon homology and cohomology essentials, but not "the" reference onalgebraic topology as a whole.
The book binding is horrible - Munkres really belabors the simplicial theory, and it getsto be quite painful (especially the*CHAPTER* on the topologicalinvariance of simplicial homology groups). - Some very important topics(homotopy theory, fiber bundles) are not at all discussed. - The bookbinding is horrible -- my copy is in two pieces, with several loose pages,and I don't think the hardcover edition is still in print.
Excellent text on homology and cohomology * point set topology (e.g. in Munkres' Topology) * Abstract algebra * Mathematicalmaturity to be willing to follow a definition and argument even when itseems like a weird side-track In addition, this would not be the firstbook I would recommend to those interested in algebraic topology.Firstmight be Massey's "Algebraic Topology: and Introduction" thatintroduces the fundamental group (conceptually easier than homology andcohomology). At some point, however, a prospective student in topologywill have to learn homological algebra and this provides the most concreteapproach I know to the subject. Algebraic topology is a lot of fun, butmany of the previous textbooks had not given that impression.This onedoes. ... Read more |
13. Differential Forms in Algebraic Topology (Graduate Texts in Mathematics) by Raoul Bott, Loring W. Tu | |
Paperback: 350
Pages
(2010-11-02)
list price: US$74.95 -- used & new: US$60.14 (price subject to change: see help) Asin: 1441928154 Average Customer Review: Canada | United Kingdom | Germany | France | Japan | |
Editorial Review Product Description Customer Reviews (7)
a masterpiece of exposition
So far so good
good book
wonderfully clear, useful book
A unique mathematics book That makes this one of my all time favourite mathematics texts. ... Read more |
14. A basic course in algebraic topology (v. 127) by W.S. Massey | |
Hardcover: 452
Pages
(1980-04-29)
list price: US$74.95 -- used & new: US$45.31 (price subject to change: see help) Asin: 038797430X Average Customer Review: Canada | United Kingdom | Germany | France | Japan | |
Editorial Review Product Description Customer Reviews (2)
Good for newbies
Excellent text on algebraic topology |
15. Homology theory: A first course in algebraic topology (Holden-Day series in mathematics) by Sze-Tsen Hu | |
Hardcover: 247
Pages
(1966)
Asin: B0006BOF7M Canada | United Kingdom | Germany | France | Japan | |
16. Lecture Notes in Algebraic Topology (Graduate Studies in Mathematics, 35) by Paul Kirk James F. Davis | |
Hardcover: 367
Pages
(2001-08-01)
list price: US$61.00 -- used & new: US$61.00 (price subject to change: see help) Asin: 0821821601 Average Customer Review: Canada | United Kingdom | Germany | France | Japan | |
Editorial Review Product Description Customer Reviews (1)
Godd for the second step. |
17. An Introduction to Algebraic Topology (Graduate Texts in Mathematics) by Joseph J. Rotman | |
Hardcover: 460
Pages
(1988-08-17)
list price: US$84.95 -- used & new: US$52.65 (price subject to change: see help) Asin: 0387966781 Average Customer Review: Canada | United Kingdom | Germany | France | Japan | |
Editorial Review Product Description Customer Reviews (3)
A readable alternative to Hatcher
Rotman does it again.
Good textbook |
18. Algebraic Topology from a Homotopical Viewpoint (Universitext) by Marcelo Aguilar, Samuel Gitler, Carlos Prieto | |
Paperback: 478
Pages
(2010-11-02)
list price: US$99.00 -- used & new: US$82.49 (price subject to change: see help) Asin: 1441930051 Canada | United Kingdom | Germany | France | Japan | |
Editorial Review Product Description The authors present introductory material in algebraic topology from a novel point of view in using a homotopy-theoretic approach. This carefully written book can be read by any student who knows some topology, providing a useful method to quickly learn this novel homotopy-theoretic point of view of algebraic topology. |
19. Algebraic Topology: Homology and Cohomology (Dover Books on Mathematics) by Andrew H. Wallace | |
Paperback: 288
Pages
(2007-10-19)
list price: US$15.95 -- used & new: US$9.39 (price subject to change: see help) Asin: 0486462390 Average Customer Review: Canada | United Kingdom | Germany | France | Japan | |
Editorial Review Product Description This self-contained treatment is suitable for advanced undergraduate and graduate students and may be used either after or concurrently with courses in general topology and algebra. It surveys several algebraic invariants: the fundamental group, singular and Cech homology groups, and a variety of cohomology groups. 1970 edition. Customer Reviews (1)
Not Specific and too Many Assumtions and Typos |
20. Algebraic Topology: An Intuitive Approach (Translations of Mathematical Monographs) by Hajime Sato | |
Paperback: 118
Pages
(1999-02)
list price: US$29.00 -- used & new: US$22.99 (price subject to change: see help) Asin: 0821810464 Average Customer Review: Canada | United Kingdom | Germany | France | Japan | |
Editorial Review Product Description In response to suggestions from readers of the original edition of this book, Sato has added an appendix of useful definitions and results on sets, general topology, groups and such. He has also provided references. Topics covered include fundamental notions such as homeomorphisms, homotopy equivalence, fundamental groups and higher homotopy groups, homology and cohomology, fiber bundles, spectral sequences and characteristic classes. Objects and examples considered in the text include the torus, the Möbius strip, the Klein bottle, closed surfaces, cell complexes and vector bundles. Customer Reviews (5)
"Intuition" more a prerequisite than a result
Excelent Start
Excellent accompaniment to Hatcher
Good Supplementary Reading The book presents the most basic ideas pertaining to homotopy, homology, cohomology, fibre bundles, spectral sequences, and characteristic classes. The emphasis is on simple examples and simple calculations to demonstrate what is going on. Rigorous definitions, proofs, and even frequently even the statements of theorems, are avoided. One good aspect of the treatment is the axiomatic presentation of homology and cohomology a la Eilenberg and Steenrod. Some of the essential material is also presented, e.g. the cup product that gives a ring structure to the cohomology group, the Kunneth theorem, the Universal Coefficient theorem, and so on. The book would afford a bird's-eye view, a conspectus, to a bright undergraduate or beginning graduate student. It goes without saying, of course, that this is for motivation, and it doesn't replace the hard technical grind required to master the subject. The book suffers in comparison to the one by Fomenko, Fuchs and Gutenmacher (Homotopic Topology), but that, alas, can't be had for love or money.
algeblaic topology |
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