Algebraic Homotopy

Algebraic Homotopy
Title Algebraic Homotopy PDF eBook
Author Hans J. Baues
Publisher Cambridge University Press
Pages 490
Release 1989-02-16
Genre Mathematics
ISBN 0521333768

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This book gives a general outlook on homotopy theory; fundamental concepts, such as homotopy groups and spectral sequences, are developed from a few axioms and are thus available in a broad variety of contexts. Many examples and applications in topology and algebra are discussed, including an introduction to rational homotopy theory in terms of both differential Lie algebras and De Rham algebras. The author describes powerful tools for homotopy classification problems, particularly for the classification of homotopy types and for the computation of the group homotopy equivalences. Applications and examples of such computations are given, including when the fundamental group is non-trivial. Moreover, the deep connection between the homotopy classification problems and the cohomology theory of small categories is demonstrated. The prerequisites of the book are few: elementary topology and algebra. Consequently, this account will be valuable for non-specialists and experts alike. It is an important supplement to the standard presentations of algebraic topology, homotopy theory, category theory and homological algebra.

Homotopy Type Theory: Univalent Foundations of Mathematics

Homotopy Type Theory: Univalent Foundations of Mathematics
Title Homotopy Type Theory: Univalent Foundations of Mathematics PDF eBook
Author
Publisher Univalent Foundations
Pages 484
Release
Genre
ISBN

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Modern Classical Homotopy Theory

Modern Classical Homotopy Theory
Title Modern Classical Homotopy Theory PDF eBook
Author Jeffrey Strom
Publisher American Mathematical Soc.
Pages 862
Release 2011-10-19
Genre Mathematics
ISBN 0821852868

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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.

Homotopy Invariant Algebraic Structures on Topological Spaces

Homotopy Invariant Algebraic Structures on Topological Spaces
Title Homotopy Invariant Algebraic Structures on Topological Spaces PDF eBook
Author J. M. Boardman
Publisher Springer
Pages 268
Release 2006-11-15
Genre Mathematics
ISBN 3540377999

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Algebraic Topology - Homotopy and Homology

Algebraic Topology - Homotopy and Homology
Title Algebraic Topology - Homotopy and Homology PDF eBook
Author Robert M. Switzer
Publisher Springer
Pages 541
Release 2017-12-01
Genre Mathematics
ISBN 3642619231

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From the reviews: "The author has attempted an ambitious and most commendable project. [...] The book contains much material that has not previously appeared in this format. The writing is clean and clear and the exposition is well motivated. [...] This book is, all in all, a very admirable work and a valuable addition to the literature." Mathematical Reviews

Rational Homotopy Theory

Rational Homotopy Theory
Title Rational Homotopy Theory PDF eBook
Author Yves Felix
Publisher Springer Science & Business Media
Pages 589
Release 2001
Genre Mathematics
ISBN 0387950680

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This is a long awaited book on rational homotopy theory which contains all the main theorems with complete proofs, and more elementary proofs for many results that were proved ten or fifteen years ago. The authors added a frist section on classical algebraic topology to make the book accessible to students with only little background in algebraic topology.

Introduction to Homotopy Theory

Introduction to Homotopy Theory
Title Introduction to Homotopy Theory PDF eBook
Author Martin Arkowitz
Publisher Springer Science & Business Media
Pages 352
Release 2011-07-25
Genre Mathematics
ISBN 144197329X

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This is a book in pure mathematics dealing with homotopy theory, one of the main branches of algebraic topology. The principal topics are as follows: Basic Homotopy; H-spaces and co-H-spaces; fibrations and cofibrations; exact sequences of homotopy sets, actions, and coactions; homotopy pushouts and pullbacks; classical theorems, including those of Serre, Hurewicz, Blakers-Massey, and Whitehead; homotopy Sets; homotopy and homology decompositions of spaces and maps; and obstruction theory. The underlying theme of the entire book is the Eckmann-Hilton duality theory. The book can be used as a text for the second semester of an advanced ungraduate or graduate algebraic topology course.