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Cohomology Operations and Applications in Homotopy Theory

Robert E. Mosher and Martin C. Tangora
Publisher: 
Dover Publications
Publication Date: 
2008
Number of Pages: 
214
Format: 
Paperback
Price: 
14.95
ISBN: 
9780486466644
Category: 
Monograph
[Reviewed by
Michelle Intermont
, on
11/9/2008
]

My very own copy of Mosher and Tangora! It arrived on my desk recently, a reprint of the 1968 text. The algebraic topology community owes a great deal of thanks to Dover for picking up this classic.

What more needs to be said? Not much! If you’re an algebraic topologist or want to be one, pick up a copy today. If, on the other hand, the words “cohomology” and “homotopy” don’t roll off your tongue, there’s no need to crack the spine here.

This is a very readable, thin volume, which does quite a bit without trying to do too much. The book provides the grounding necessary for the methods discussed, while refraining from excruciating detail. For example, before the Serre spectral sequence is used, there is a concise set up of spectral sequences and fibre spaces. This approach lends greatly to the readability of the text. The particular cohomology operations known as the Steenrod squares form the backbone of this book, but unlike Steenrod and Epstein’s earlier book on cohomology operations, Eilenberg-MacLane spaces are also prominent here. The Adams spectral sequence and some calculuations regarding the stable homotopy groups of spheres are also nicely presented.


Michele Intermont is an associate professor of mathematics at Kalamazoo College in Kalamazoo, MI.

Preface
1. Introduction to cohomology operations
2. Construction of the Steenrod squares
3. Properties of the squares
4. Application: the Hopf invariant
5. Application: vector fields on spheres
6. The Steenrod algebra
7. Exact couples and spectral sequences
8. Fibre spaces
9. Cohomology of K(pi, n)
10. Classes of Abelian groups
11. More about fiber spaces
12. Applications: some homotopy groups of spheres
13. n-Type and Postnikov systems
14. Mapping sequences and homotopy classification
15. Properties of the stable range
16. Higher cohomology operations
17. Compositions in the stable homotopy of spheres
18. The Adams spectral sequence
Bibliography
Index

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