Algebraic Topology

Rating:
Summary: Probably better as a 2nd (or 3rd) course rather than 1st
Review: Most mathematicians, I suspect, can relate to the "colloquium experience": the first minutes of a lecture go easily, followed by twenty or thirty of real edification, concluded by ten to fifteen of feeling lost. I regret to say that this was pretty much my experience with the book. Fulton writes with unusual enthusiasm and the first two- thirds of the book is a joy to read, even while it is real work. I imagine that he must be a remarkable teacher in person. He has some threads such as winding numbers and the Mayer-Vietoris Sequence that continue throughout the book, bringing unity to a wide selection of topics. There are a number of applications of the subject to other areas, such as complex analysis (Riemann surfaces) and algebraic geometry (the Riemann-Roch Theorem), to name only two. There are particularly interesting illustrations of the Brouwer Fixed Point Theorem and related results. Unfortunately, there are two rather major reservations I have about the book. The first, already alluded to, is that it seemed to me to become precipitously difficult towards the end. The second is that this book would be excellent for a second or perhaps third course in the subject rather than a first. While the topics he covers are interesting in their own right, I still favor a more "standard" approach covering simplicial complexes, homology, CW complexes, and homotopy theory with higher homotopy groups, such as in the books by Maunder, Munkres, or Rotman (the last two of which I recommend unreservedly). It is true that Fulton has some coverage these topics, and a particularly extensive discussion of group actions and G-spaces, but he presupposes a background or ability that the novice to algebraic topology is unlikely to have. I would like to recommend this book, as I found it very edifying, but it seems better suited for one with some prior acquaintance to the subject.

Rating:
Summary: A book of ideas
Review: This book is an introduction to algebraic topology that is written by a master expositor. Many books on algebraic topology are written much too formally, and this makes the subject difficult to learn for students or maybe physicists who need insight, and not just functorial constructions, in order to learn or apply the subject. Anyone learning mathematics, and especially algebraic topology, must of course be expected to put careful thought into the task of learning. However, it does help to have diagrams, pictures, and a certain degree of handwaving to more greatly appreciate this subject.

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.

Rating:
Summary: This is one of the great algebraic topology books!
Review: This is a book for people who want to think about topology, not just learn a lot of fancy definitions and then mechanically compute things. Fulton has put the essence of Algebraic Topology into this book, much in the way Mike Artin has done with his "Algebra". In my opinion, he should win some sort of expository award for it.

Rating:
Summary: This is one of the great algebraic topology books!
Review: This is a book for people who want to think about topology, not just learn a lot of fancy definitions and then mechanically compute things. Fulton has put the essence of Algebraic Topology into this book, much in the way Mike Artin has done with his "Algebra". In my opinion, he should win some sort of expository award for it.