**Translations of Mathematical Monographs**

2001;
235 pp;
Hardcover

MSC: Primary 55;

Print ISBN: 978-0-8218-2170-1

Product Code: MMONO/198

List Price: $103.00

AMS Member Price: $82.40

MAA Member Price: $92.70

**Electronic ISBN: 978-1-4704-4624-6
Product Code: MMONO/198.E**

List Price: $103.00

AMS Member Price: $82.40

MAA Member Price: $92.70

# Simplicial and Operad Methods in Algebraic Topology

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*V. A. Smirnov*

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.

#### Readership

Graduate students and research mathematicians working in algebraic topology.

#### Table of Contents

# Table of Contents

## Simplicial and Operad Methods in Algebraic Topology

- Cover Cover11
- Title page v6
- Copyright page vi7
- Contents vii8
- Preface ix10
- Operads in the category of topological spaces 112
- Simplicial objects and homotopy theory 1930
- Algebraic structures on chain complexes 4556
- 𝐴_{∞}-structures on chain complexes 7586
- Operads and algebras over operads 89100
- Homology of iterated loop spaces 119130
- Homotopy theories and 𝐸_{∞}-structures 133144
- Operad methods in cobordism theory 157168
- Description of the cohomology of groups and algebras 183194
- Homology operations and differentials in the Adams spectral sequence 209220
- Bibliography 231242
- Index 233244
- Back Cover Back Cover1249