Secondary Cohomology Operations

by Harper, John R.

• ISBN13: 9780821831984
• ISBN10: 0821831984
• Format: Hardcover
• Copyright: 2002-07-01
• Publisher: Amer Mathematical Society

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