9780821831984

Secondary Cohomology Operations

by
  • ISBN13:

    9780821831984

  • ISBN10:

    0821831984

  • Format: Hardcover
  • Copyright: 2002-07-01
  • Publisher: Amer Mathematical Society

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Summary

Although the theory and applications of secondary cohomology operations are an important part of an advanced graduate-level algebraic topology course, there are few books on the subject. The AMS now fills that gap with the publication of the present volume. The author's main purpose in this book is to develop the theory of secondary cohomology operations for singular cohomology theory, which is treated in terms of elementary constructions from general homotopy theory. Among manyapplications considered are the Hopf invariant one theorem (for all primes $p$, including $p = 2$), Browder's theorem on higher Bockstein operations, and cohomology theory of Massey-Peterson fibrations. Numerous examples and exercises help readers to gain a working knowledge of the theory. A summary ofmore advanced parts of the core material is included in the first chapter. Prerequisite is basic algebraic topology, including the Steenrod operations. The book is geared toward graduate students and research mathematicians interested in algebraic topology and can be used for self-study or as a textbook for an advanced course on the topic. It is available in both hardcover and softcover editions.

Table of Contents

Preface vii
Review of Primary Operations
1(40)
Primary cohomology operations
1(3)
Steenrod operations and the Steenrod algebra
4(17)
Cohomology operations and Eilenberg-Mac Lane spaces
21(8)
Steenrod operations and maps between spheres
29(4)
Steenrod operations for odd primes
33(8)
Segue to Secondary Operations
41(4)
Fundamental Constructions
45(40)
Terminology and conventions
45(4)
Basic constructions
49(11)
Dependence on homotopies
60(4)
Peterson-Stein formulas
64(5)
The Milnor filtration
69(16)
Secondary Cohomology Operations
85(44)
Definitions
85(5)
The fundamental theorem for secondary operations
90(31)
Higher order operations
121(8)
Calculations with Secondary Operations
129(50)
Operations based on Sq(2)Sq(n)
129(9)
Higher order Bocksteins
138(9)
The Adams Operations
147(8)
The Liulevicius-Shimada-Yamanashita operations
155(5)
Operations based on unstable relations
160(19)
The Hopf Invariant
179(20)
The classical Hopf invariant
179(9)
Hopf invariant one
188(11)
The Cohomology Structure of Universal Examples
199(62)
Unstable modules and algebras over the Steenrod algebra
199(6)
Massey-Peterson fibrations
205(24)
Products
229(32)
Bibliography 261(6)
Index 267

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