Modern Classical Homotopy Theory

Jeffrey Strom

• Publisher: American Mathematical Society
• Year: 2011
• Print ISBN:
  9781470471637
• eBook ISBN:
  9781470411886
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

The 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.

Table of contents

• Cover
• Title page
• Contents
• Preface
• Part I. The language of categories
• Categories and functors
• Limits and colimits
• Part II. 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
• Part III. Four topological inputs
• The concept of dimension in homotopy theory
• Subdivision of disks
• The local nature of fibrations
• Pullbacks of cofibrations
• Related topics
• Part IV. Targets as domains, domains as targets
• Constructions of spaces and maps
• Understanding suspension
• Comparing pushouts and pullbacks
• Some computations in homotopy theory
• Further topics
• Part V. Cohomology and homology
• Cohomology
• Homology
• Cohomology operations
• Chain complexes
• Topics, problems and projects
• Part VI. 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
• Part VII. Vistas
• Localization and completion
• Exponents for homotopy groups
• Classes of spaces
• Miller’s theorem
• Some algebra
• References
• Index of notation
• Index
• Back Cover