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Modern Classical Homotopy Theory

Jeffrey Strom (Autor)

Buch | Softcover

835 Seiten

2011
American Mathematical Society (Verlag)
978-1-4704-7163-7 (ISBN)

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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.

Jeffrey Strom, Western Michigan University, Kalamazoo, MI.

The 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
Index

Erscheinungsdatum	13.02.2023
Reihe/Serie	Graduate Studies in Mathematics
Verlagsort	Providence
Sprache	englisch
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
ISBN-10	1-4704-7163-9 / 1470471639
ISBN-13	978-1-4704-7163-7 / 9781470471637
Zustand	Neuware