Algebraic Homotopy

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Cambridge University Press, Feb 16, 1989 - Mathematics - 466 pages
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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.
 

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Contents

I Axioms for homotopy theory and examples of eofibration categories
1
II Homotopy theory in a cofibration category
83
III The homotopy spectral sequences in a cofibration category
179
IV Extensions coverings and cohomology groups of a category
229
V Maps between mapping cones
258
VI Homotopy theory of CWcompIexes
306
VII Homotopy theory of complexes in a cofibration category
371
VIII Homotopy theory of Postnikov towers and the Sullivande Rham equivalence of rational homotopy categories ...
404
IX Homotopy theory of reduced complexes
436
Bibliography
455
Index
461
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