**Translations of Mathematical Monographs
Iwanami Series in Modern Mathematics**

2006; 254 pp; Softcover

MSC: Primary 55;

**Print ISBN: 978-0-8218-3514-2**

Product Code: MMONO/230

Product Code: MMONO/230

List Price: $53.00

AMS Member Price: $42.40

MAA Member Price: $47.70

**Electronic ISBN: 978-1-4704-4788-5
Product Code: MMONO/230.E**

List Price: $53.00

AMS Member Price: $42.40

MAA Member Price: $47.70

#### Supplemental Materials

# Generalized Cohomology

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*Akira Kono; Dai Tamaki*

In the 1950s, Eilenberg and Steenrod presented their
famous characterization of homology theory by seven axioms. Somewhat later, it
was found that keeping just the first six of these axioms (all except the
condition on the “homology” of the point), one can obtain many
other interesting systems of algebraic invariants of topological manifolds,
such as \(K\)-theory, cobordisms, and others. These theories come under
the common name of generalized homology (or cohomology) theories.

The purpose of the book is to give an exposition of generalized (co)homology
theories that can be read by a wide group of mathematicians who are not experts
in algebraic topology. It starts with basic notions of homotopy theory and then
introduces the axioms of generalized (co)homology theory. Then the authors
discuss various types of generalized cohomology theories, such as
complex-oriented cohomology theories and Chern classes, \(K\)-theory,
complex cobordisms, and formal group laws. A separate chapter is devoted to
spectral sequences and their use in generalized cohomology theories.

The book is intended to serve as an introduction to the subject for
mathematicians who do not have advanced knowledge of algebraic topology.
Prerequisites include standard graduate courses in algebra and topology, with
some knowledge of ordinary homology theory and homotopy theory.

#### Readership

Graduate students and research mathematicians interested in algebraic topology.

#### Reviews & Endorsements

The book is a successful guide because it provides a smooth path from basics into the deeper parts of complex \(K\)-theory and complex cobordism theory. It supplies details for some crucial theorems and directs the reader to excellent treatments of quoted material. ...the book provides students and other newcomers with the language needed to converse with an expert.

-- Bulletin of the American Mathematical Society

#### Table of Contents

# Table of Contents

## Generalized Cohomology

- Cover Cover11
- Title page iii4
- Contents v6
- Preface to the english edition vii8
- Preface ix10
- Preliminaries 114
- Generalized cohomology 1932
- Characteristic classes of vector bundles 4356
- 𝐾-theory 6780
- The Spectral sequence 115128
- Complex cobordism and its applications 179192
- Appendix A. Simplicial techniques 205218
- Appendix B. Limits 217230
- Appendix C. Spectrum 225238
- Bibliography 241254
- Index 251264
- Back Cover Back Cover1270