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Introduction to Homotopy Theory

Martin Arkowitz
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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Arkowitz’ Introduction to Homotopy Theory is presumably aimed at an audience of graduate students who have already been exposed to the basics of algebraic topology, viz. homology and cohomology and the fundamental group, in a first course in the subject. To be sure, the book’s appendices include material on these topics in a pro forma fashion and reminders about general themes in homological algebra besides. But it’s clear that these are indeed just meant as reminders and a vade mecum of useful material from the past. The reader, or the student, should be well-versed in this background material, for, although Arkowitz does take great care to aim his discussion at “students [wishing] to learn the material… [whence] the proofs in the text contain a great deal of detail…,” he covers a good deal of ground and requires his charges to engage the material very seriously. To wit, the book is anything but chatty, and comes supplied with a large number of exercises. And the frequently used protocol of designating certain of these as “more difficult” or as “[to be] used in the text” is in place, underscoring the fact that thorough mastery of this material involves doing a lot of serious homework — lest, for instance, the flow of later material be compromised. This is clearly a very sound pedagogical approach, but it certainly calls for non-trivial student commitment.

Another noteworthy pedagogical fact about Introduction to Homotopy Theory is the author’s choice to base much of his development on Eckmann-Hilton duality theory. Says Arkowitz, “The Eckmann-Hilton theory has been around for about fifty years but there appears to be no book-length exposition of it, apart from the early lecture notes of Hilton.” Thus, the book under review aims, if not to break new ground, then to right a pedagogical wrong: “Dual concepts occur in pairs, such as H-space and co-H-space, fibration and cofibration, loop space and suspension, … and so do many theorems. We often give complete details in describing one of these and only sketch its dual … when the latter can essentially be derived by dualization …” Given the ubiquity and fecundity of the very notion of duality in so many parts of modern mathematics, with algebraic topology in the shadows if not the spot-light, this teaching tactic is clearly very sound. It contributes to the pupil’s development into a creative worker at a relatively early stage.

Introduction to Homotopy Theory is presented in nine chapters, taking the reader from “basic homotopy” to obstruction theory with a lot of marvelous material in between: the dual concepts mentioned above are well-represented, of course, as are, e.g., the “Theorems of Serre, Hurewicz, and Blakers-Massey.” There is the obligatory coverage of universal coefficient theorems, of course, and the business of calculation of homotopy groups. Eckmann-Hilton duality properly so-called is covered in two phases, first on p.67ff., right after Eilenberg-Mac Lane spaces and Moore spaces, which are regarded as “dual to each other in the weak sense,” and then on p.225 ff., right after Arkowitz’ discussion of the three name-carrying theorems just mentioned. This transit into deeper waters includes a reality-check of sorts: on p. 226 we read that “[t]he dual of a proven result is sometimes false. We give four examples of this …” Ah, yes, the idiosyncratic quality of topology as a means whereby to instill the scholarly virtue of prudence is clearly in evidence: don’t let loose intuition and wishful thinking replace the nuts and bolts of very careful theorem proving and possibly searching for a Gegenbeispiel.

Arkowitz’ book is a valuable text and promises to figure prominently in the education of many young topologists.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

1 Basic Homotopy.- 1.1 Introduction.- 1.2 Spaces, Maps, Products and Wedges.- 1.3 Homotopy I.- 1.4 Homotopy II.- 1.5 CW Complexes.- 1.6 Why Study Homotopy Theory?.- Exercises.- 2 H-Spaces and Co-H-Spaces.- 2.1 Introduction.- 2.2. H-Spaces and Co-H-Spaces.- 2.3 Loop Spaces and Suspensions.- 2.4 Homotopy Groups I.- 2.5 Moore Spaces and Eilenberg-Mac Lane Spaces.- 2.6 Eckmann-Hilton Duality I.- Exercises.- 3 Cofibrations and Fibrations.- 3.1 Introduction.- 3.2 Cofibrations.- 3.3 Fibrations.- 3.4 Examples of Fiber Bundles.- 3.5 Replacing a Map by a Cofiber or Fiber Map.- Exercises.- 4 Exact Sequences.- 4.1 Introduction.- 4.2 The Coexact and Exact Sequence of a Map.- 4.3 Actions and Coactions.- 4.4 Operations.- 4.5 Homotopy Groups II.- Exercises.- 5 Applications of Exactness.- 5.1 Introduction.- 5.2 Universal Coefficient Theorems.- 5.3 Homotopical Cohomology Groups.- 5.4 Applications to Fiber and Cofiber Sequences.- 5.5 The Operation of the Fundamental Group.- 5.6 Calculation of Homotopy Groups.-Exercises.- 6 Homotopy Pushouts and Pullbacks.- 6.1 Introduction.- 6.2 Homotopy Pushouts and Pullbacks I.- 6.3 Homotopy Pushouts and Pullbacks II.- 6.4 Theorems of Serre, Hurewicz and Blakers-Massey.- 6.5 Eckmann-Hilton Duality II.- Exercises.- 7 Homotopy and Homology Decompositions.- 7.1 Introduction.- 7.2 Homotopy Decompositions of Spaces.- 7.3 Homology Decompositions of Spaces.- 7.4 Homotopy and Homology Decompositions of Maps.- Exercises.- 8 Homotopy Sets.- 8.1 Introduction.- 8.2 The Set [X, Y].- 8.3 Category.- 8.4 Loop and Group Structure in [X, Y].-Exercises.- 9 Obstruction Theory.- 9.1 Introduction.- 9.2 Obstructions Using Homotopy Decompositions.- 9.3 Lifts and Extensions.- 9.4 Obstruction Miscellany.- Exercises.- A Point-Set Topology.- B The Fundamental Group.- C Homology and Cohomology.- D Homotopy Groups of the n-Sphere.- E Homotopy Pushouts and Pullbacks.- F Categories and Functors.- Hints to Some of the Exercises.- References.- Index.-