Nilpotence and periodicity in stable homotopy theory 1st Edition by Douglas Ravenel 069108792X 9780691087924

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ISBN 10: 069108792X 

ISBN 13: 9780691087924

Author: Douglas C. Ravenel

Nilpotence and Periodicity in Stable Homotopy Theory describes some major advances made in algebraic topology in recent years, centering on the nilpotence and periodicity theorems, which were conjectured by the author in 1977 and proved by Devinatz, Hopkins, and Smith in 1985. During the last ten years a number of significant advances have been made in homotopy theory, and this book fills a real need for an up-to-date text on that topic. Ravenel’s first few chapters are written with a general mathematical audience in mind. They survey both the ideas that lead up to the theorems and their applications to homotopy theory. The book begins with some elementary concepts of homotopy theory that are needed to state the problem. This includes such notions as homotopy, homotopy equivalence, CW-complex, and suspension. Next the machinery of complex cobordism, Morava K-theory, and formal group laws in characteristic p are introduced. The latter portion of the book provides specialists with a coherent and rigorous account of the proofs. It includes hitherto unpublished material on the smash product and chromatic convergence theorems and on modular representations of the symmetric group.

Table of contents:

1 The main theorems

1.1 Homotopy

1.2 Functors

1.3 Suspension

1.4 Self-maps and the nilpotence theorem

1.5 Morava K-theories and the periodicity theorem

2 Homotopy groups and the chromatic filtration

2.1 The definition of homotopy groups

2.2 Classical theorems

2.3 Cofibres

2.4 Motivating examples

2.5 The chromatic filtration

3 MU-theory and formal group laws

3.1 Complex bordism

3.2 Formal group laws

3.3 The category CΓ

3.4 Thick subcategories

4 Morava’s orbit picture and Morava stabilizer groups

4.1 The action of Γ on L

4.2 Morava stabilizer groups

4.3 Cohomological properties of Sn

5 The thick sub category theorem

5.1 Spectra

5.2 Spanier-Whitehead duality

5.3 The proof of the thick subcategory theorem

6 The periodicity theorem

6.1 Properties of vn-maps

6.2 The Steenrod algebra and Margolis homology groups

6.3 The Adams spectral sequence and the vn-map on Y

6.4 The Smith construction

7 Bousfield localization and equivalence

7.1 Basic definitions and examples

7.2 Bousfield equivalence

7.3 The structure of (MU)

7.4 Some classes bigger than (MU)

7.5 E(n)-localization and the chromatic filtration

8 The proofs of the localization, smash product and chromatic convergence theorems

8.1 LnBP and the localization theorem

8.2 Reducing the smash product theorem to a special example

8.3 Constructing a finite torsion free prenilpotent spectrum

8.4 Some cohomological properties of profinite groups

8.5 The action of Sm on FK(m)*(CP^∞)

8.6 Chromatic convergence

9 The proof of the nilpotence theorem

9.1 The spectra X(n)

9.2 The proofs of the first two lemmas

9.3 A paradigm for proving the third lemma

9.4 The Snaith splitting of Ω^S^2m+1

9.5 The proof of the third lemma

9.6 Historical note: theorems of Nishida and Toda

A Some tools from homotopy theory

A.1 CW-complexes

A.2 Loop spaces and spectra

A.3 Generalized homology and cohomology theories

A.4 Brown representability

A.5 Limits in the stable homotopy category

A.6 The Adams spectral sequence

B Complex bordism and BP-theory

B.1 Vector bundles and Thom spectra

B.2 The Pontrjagin-Thom construction

B.3 Hopf algebroids

B.4 The structure of MU* (MU)

B.5 BP-theory

B.6 The Landweber exact functor theorem

B.7 Morava K-theories

B.8 The change-of-rings isomorphism and the chromatic spectral sequence

C Some idempotents associated with the symmetric group

C.1 Constructing the idempotents

C.2 Idempotents for graded vector spaces

C.3 Getting strongly type n spectra from partially type n spectra

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