Hardcover ISBN:  9780821821701 
Product Code:  MMONO/198 
List Price:  $165.00 
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AMS Member Price:  $132.00 
eBook ISBN:  9781470446246 
Product Code:  MMONO/198.E 
List Price:  $155.00 
MAA Member Price:  $139.50 
AMS Member Price:  $124.00 
Hardcover ISBN:  9780821821701 
eBook: ISBN:  9781470446246 
Product Code:  MMONO/198.B 
List Price:  $320.00 $242.50 
MAA Member Price:  $288.00 $218.25 
AMS Member Price:  $256.00 $194.00 
Hardcover ISBN:  9780821821701 
Product Code:  MMONO/198 
List Price:  $165.00 
MAA Member Price:  $148.50 
AMS Member Price:  $132.00 
eBook ISBN:  9781470446246 
Product Code:  MMONO/198.E 
List Price:  $155.00 
MAA Member Price:  $139.50 
AMS Member Price:  $124.00 
Hardcover ISBN:  9780821821701 
eBook ISBN:  9781470446246 
Product Code:  MMONO/198.B 
List Price:  $320.00 $242.50 
MAA Member Price:  $288.00 $218.25 
AMS Member Price:  $256.00 $194.00 

Book DetailsTranslations of Mathematical MonographsVolume: 198; 2001; 235 ppMSC: Primary 55;
In recent years, for solving problems of algebraic topology and, in particular, difficult problems of homotopy theory, algebraic structures more complicated than just a topological monoid, an algebra, a coalgebra, etc., have been used more and more often. A convenient language for describing various structures arising naturally on topological spaces and on their cohomology and homotopy groups is the language of operads and algebras over an operad. This language was proposed by J. P. May in the 1970s to describe the structures on various loop spaces.
This book presents a detailed study of the concept of an operad in the categories of topological spaces and of chain complexes. The notions of an algebra and a coalgebra over an operad are introduced, and their properties are investigated. The algebraic structure of the singular chain complex of a topological space is explained, and it is shown how the problem of homotopy classification of topological spaces can be solved using this structure. For algebras and coalgebras over operads, standard constructions are defined, particularly the bar and cobar constructions. Operad methods are applied to computing the homology of iterated loop spaces, investigating the algebraic structure of generalized cohomology theories, describing cohomology of groups and algebras, computing differential in the Adams spectral sequence for the homotopy groups of the spheres, and some other problems.
ReadershipGraduate students and research mathematicians working in algebraic topology.

Table of Contents

Chapters

Operads in the category of topological spaces

Simplicial objects and homotopy theory

Algebraic structures on chain complexes

$A_\infty $structures on chain complexes

Operads and algebras over operads

Homology of iterated loop spaces

Homotopy theories and $E_\infty $structures

Operad methods in cobordism theory

Description of the cohomology of groups and algebras

Homology operations and differentials in the Adams spectral sequence


Additional Material

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In recent years, for solving problems of algebraic topology and, in particular, difficult problems of homotopy theory, algebraic structures more complicated than just a topological monoid, an algebra, a coalgebra, etc., have been used more and more often. A convenient language for describing various structures arising naturally on topological spaces and on their cohomology and homotopy groups is the language of operads and algebras over an operad. This language was proposed by J. P. May in the 1970s to describe the structures on various loop spaces.
This book presents a detailed study of the concept of an operad in the categories of topological spaces and of chain complexes. The notions of an algebra and a coalgebra over an operad are introduced, and their properties are investigated. The algebraic structure of the singular chain complex of a topological space is explained, and it is shown how the problem of homotopy classification of topological spaces can be solved using this structure. For algebras and coalgebras over operads, standard constructions are defined, particularly the bar and cobar constructions. Operad methods are applied to computing the homology of iterated loop spaces, investigating the algebraic structure of generalized cohomology theories, describing cohomology of groups and algebras, computing differential in the Adams spectral sequence for the homotopy groups of the spheres, and some other problems.
Graduate students and research mathematicians working in algebraic topology.

Chapters

Operads in the category of topological spaces

Simplicial objects and homotopy theory

Algebraic structures on chain complexes

$A_\infty $structures on chain complexes

Operads and algebras over operads

Homology of iterated loop spaces

Homotopy theories and $E_\infty $structures

Operad methods in cobordism theory

Description of the cohomology of groups and algebras

Homology operations and differentials in the Adams spectral sequence