Elements of Homotopy Theory (Graduate Texts in Mathematics, 61) by George W. Whitehead (PDF)

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As the title suggests, this book is concerned with the elementary portion of the subject of homotopy theory. It is assumed that the reader is familiar with the fundamental group and with singular homology theory, including the Universal Coefficient and Kiinneth Theorems. Some acquaintance with manifolds and Poincare duality is desirable, but not essential. Anyone who has taught a course in algebraic topology is familiar with the fact that a formidable amount of technical machinery must be introduced and mastered before the simplest applications can be made. This phenomenon is also observable in the more advanced parts of the subject. I have attempted to short-circuit it by making maximal use of elementary methods. This approach entails a leisurely exposition in which brevity and perhaps elegance are sacrificed in favor of concreteness and ease of application. It is my hope that this approach will make homotopy theory accessible to workers in a wide range of other subjects-subjects in which its impact is beginning to be felt. It is a consequence of this approach that the order of development is to a certain extent historical. Indeed, if the order in which the results presented here does not strictly correspond to that in which they were discovered, it nevertheless does correspond to an order in which they might have been discovered had those of us who were working in the area been a little more perspicacious.

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⭐Defining homotopy and homotopy groups is fairly easy, but their actual calculation or determination is very difficult in general. This book, written by one of the major contributors of the subject, is comprehensive and can serve as both a textbook and as a reference since there are collections of exercises at the end of each chapter. The author also includes some brief historical background from time to time, making the reading much more palatable. Space forbids a detailed review, so only the main points of each chapter will be discussed.Chapter 1: Elementary notions of homotopy are discussed, with emphasis on map extension and maps of the n-sphere into itself. Compactly generated spaces are introduced and all the spaces in the book are assumed to be compactly generated. The homotopy lifting property is used to motivate the concept of a fibration. Unfortunately the Hurewicz theorem on fibrations over paracompact spaces is not proven. The proof involves partitions of unity and is very illustrative of typical methods. Best part of the chapter: the discussion on the duality between a fibration and a cofibration.Chapter 2: This is a review of CW-complexes, which are introduced as the easiest objects to study in algebraic topology. The author covers the homology and cohomology theory of CW-complexes cellular maps. The best part of the chapter though is on products and the cohomology ring: the author explains in detail and gives special insight as to why calculation of cup products is difficult for a general CW-complex (there is no Alexander-Cech-Whitney formula for doing this, as is the case for simplicial complexes).Chapter 3: The author considers the task of putting a natural product on the set of homotopy classes of mappings. This results in a discussion of H- and H’-spaces and eventually their homology. The fundamental group makes its appearance here, along with Hopf algebras, the latter being very important recently in the context of “quantum groups”. Best part of the chapter: the discussion on the need for the suspension and smash products to alleviate the dependence on base points.Chapter 4: The author concentrates on homotopy groups in more detail, asking first to what extent homotopy can be viewed as having the same properties as homology. Defining first relative homotopy groups and the higher homotopy groups, the connection between homotopy and homology is eventually done via the Hurewicz map and the Whitehead theorem. The best part of the chapter is on the difference between homology groups and homotopy groups under cofibrations, with the opposite occurring for fibrations.Chapter 5: The author returns to the study of CW-complexes, where he studies their (relative) homotopy groups. The relative homotopy group for n greater than or equal to 3 is calculated and shown to be a free Z-module over the first homotopy group of the subcomplex with one basis element for each n-cell, in analogy to the homology of CW-complexes, wherein the nth homology group is free abelian with one basis element for each n-cell of the pair. Eilenberg-Maclane spaces are introduced here for the first time. By far the best part of the chapter though is the treatment of obstruction theory.Chapter 6: Because of its importance to the theory of characteristic classes, and because it is a nice overview of what the topologist Norman Steenrod began in his book on fiber bundles, everything in this chapter is interesting. It is shown in detail how to assign to every pair a homology group with local coefficients. Since characteristic classes are so very important in topology, and since their treatment in the literature is usually too formal, this chapter is a real treat, as it offers rare insight into the origins and intuition behind characteristic classes.Chapter 7: The homology groups of a fibration are discussed, with attention initially made to those whose base space is a suspension of another space. This motivates the James reduced products for the loop space of a suspension. The best part though is the detailed treatment of fibrations with spherical fiber.Chapter 8: The homology groups of the loop space of a space B are related to those of B via the homology suspension. This discussion leads to the best part of the chapter: the discussion of stable operations and Steenrod squares.Chapter 9: Eilenberg-Maclane spaces are used build a space of given homotopy type. The maps from one component to one of lower dimension leads to the Postnikov invariant of the space and then Postnikov systems, which is the best part of the chapter.Chapter 10: The Lusternik-Schnirelmann category arises here, and used to prove nilpotency of homotopy mappings of a space to a group. The best part is the one on the Whitehead product.Chapter 11: Homotopy operations are treated in detail, being motivated by the desire to emulate the ability of Eilenberg-MacLane spaces to give universal examples of cohomology operations. The proof of the Hilton-Milnor theorem, which gives a connection between the homotopy groups of the wedge of two spaces and the homotopy of the spaces themselves, is the best part of the chapter.Chapter 12: Stable homotopy groups are discussed and shown to behave like homology groups, at least if the dimension axiom is relaxed. In that regard the discussion on the comparison with the Eilenberg-Steenrod axioms is the best part of the chapter.Chapter 13: This is the most difficult of all the chapters in the book, for it introduces the method of spectral sequences for studying the homology of fibrations. Closely connected with later work on K-theory, spectral sequences are give an elegant treatment here, making this chapter also one of the best in the book. The author shows in detail how to associate with the fiber map an exact couple whose spectral sequence connects the homology of the base with local coefficients in the fiber, to the homology of the total space.

⭐This tome should really serve only as a reference work for research students and their mentors.It is difficult to learn algebraic topology solely from this book. Exercises (rather, merely a collection of unproven results relegated to the exercises) are scattered throughout, albeit difficult, but even so; with the existing mass-produced student-friendly alg top books on the market now (some of which are actually quite good) one would do well, or perhaps better, to scope those out first. And for those who like to relish their mathematics, one could then go on to pick up Whitehead accordingly.Given its incredible size, it makes an honest attempt at being encyclopaedic, and it gives off somewhat of a terse and dry exposition of the material at best. (I should remark that, there are some inserts where the author’s insipid sense of humour tries to break through, but this isn’t always altogether comforting!) It can still be a fun read, if you at least have a nodding acquaintance with homotopy and homology (as this book certainly goes well beyond the scope of a second/third course in topology)…. and whatever happened to the sequel, as promised by the author in the preface?Happy reading!-A

⭐Toujours une référence !

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⭐本書は、ホモトピー論の本格的な教科書である。予備知識として、ホモロジー・コホモロジー理論の基礎を一通り仮定しているが、読者は J. Munkres 著「elements of algebraic topology」を読んでおけば事足りるであろう。但し、記述の面で注意することが、一つだけある。Munkres の本における cohomology cross product と 本書によるものとでは、同じ記号 f×g を使っているが、その定義は、符号が異なるのである： f が p 次元 cochain、g が q 次元 cochain のとき、(-1)^{pq} だけ、符号が異なっている。しかしながら、本書の行間は非常に広い。特に、第６章の local homology の公理の証明に、読者は、多大な労力を払わなければならないであろう。さらに、weak homotopy equivalence が、local homology (cohomology) の同型を誘導するという性質も、暗に仮定している。この辺りが、この本の難関であろう。また、本書の著しい特徴として、全ての議論を k-space の圏で行っていることが挙げられる。(本書の記述を、通常の位相空間の圏の場合に書き直すのは、良い演習問題になるであろう。) その上で、第１章〜第５章まででは、ホモトピー集合の基本的な性質を、ホモロジー理論と関連付けながら述べている。殊に、long exact homotopy sequence に関する議論は、本書以上の美しさ・明快さを、他の文献では、見られないであろう。第６章が先に述べた local homology の理論と障害理論である。第７章から、最後の第１３章までは、ホモトピー理論の最前線で、必要不可欠とも言える道具を、いくつも紹介している。Wang 列、コホモロジー作用素、ポストニコフ分解、Samelson product、Whitehead product、EHP 列、Hopf invariant、安定ホモトピー、スペクトル系列などである。特に、第１３章のスペクトル系列の理論は、圧巻である。一般（コ）ホモロジーに対して、Atiyah-Hirzebruch のスペクトル系列をexact couple を用いて定式化し、その特別の場合として、特異（コ）ホモロジーに対する Leray-Serre のスペクトル系列（コホモロジー環の積構造含む）の理論を述べる。特に、後者の応用として、一番最後の定理として、Serre の学位論文の main theorem が紹介されている。

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