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OverviewThe core of classical homotopy theory is a body of ideas and theorems that emerged in the 1950s and was later largely codified in the notion of a model category. This core includes the notions of fibration and cofibration; CW complexes; long fiber and cofiber sequences; loop spaces and suspensions; and so on. Brown's representability theorems show that homology and cohomology are also contained in classical homotopy theory. This text develops classical homotopy theory from a modern point of view, meaning that the exposition is informed by the theory of model categories and that homotopy limits and colimits play central roles. The exposition is guided by the principle that it is generally preferable to prove topological results using topology (rather than algebra). The language and basic theory of homotopy limits and colimits make it possible to penetrate deep into the subject with just the rudiments of algebra. The text does reach advanced territory, including the Steenrod algebra, Bott periodicity, localization, the Exponent Theorem of Cohen, Moore, and Neisendorfer, and Miller's Theorem on the Sullivan Conjecture. Thus the reader is given the tools needed to understand and participate in research at (part of) the current frontier of homotopy theory. Proofs are not provided outright. Rather, they are presented in the form of directed problem sets. To the expert, these read as terse proofs; to novices they are challenges that draw them in and help them to thoroughly understand the arguments. Full Product DetailsAuthor: Jeffrey StromPublisher: American Mathematical Society Imprint: American Mathematical Society ISBN: 9781470471637ISBN 10: 1470471639 Pages: 835 Publication Date: 30 January 2011 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: In Print  This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of ContentsThe language of categories: Categories and functors Limits and colimits Semi-formal homotopy theory: Categories of spaces Homotopy Cofibrations and fibrations Homotopy limits and colimits Homotopy pushout and pullback squares Tools and techniques Topics and examples Model categories Four topological inputs: The concept of dimension in homotopy theory Subdivision of disks The local nature of fibrations Pullbacks of cofibrations Related topics Targets as domains, domains as targets: Constructions of spaces and maps Understanding suspension Comparing pushouts and pullbacks Some computations in homotopy theory Further topics Cohomology and homology: Cohomology Homology Cohomology operations Chain complexes Topics, problems and projects Cohomology, homology and fibrations: The Wang sequence Cohomology of filtered spaces The Serre filtration of a fibration Application: Incompressibility The spectral sequence of a filtered space The Leray-Serre spectral sequence Application: Bott periodicity Using the Leray-Serre spectral sequence Vistas: Localization and completion Exponents for homotopy groups Classes of spaces Miller's theorem Some algebra References Index of notation IndexReviewsObviously the book was a labor of love for its author: this is visible on every page. The coverage of the material is, in a word, amazing, even to an outsider like me. The book is well-written, as I have already indicated, and Strom's problems first-and-foremost approach is bound to be a big pedagogical hit for those who can handle it, both in front of the class and in it. The book under review is a wonderful contribution indeed. - MAA Reviews Author InformationJeffrey Strom, Western Michigan University, Kalamazoo, MI. Tab Content 6Author Website:Countries AvailableAll regions |
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