Complex Cobordism and Stable Homotopy Groups of Spheres - Ravenel.pdf

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Complex Cobordism and
Stable Homotopy Groups of Spheres
Douglas C. Ravenel
Department of Mathematics, University of Rochester, Rochester,
New York
To my wife, Michelle
Contents
List of Figures
List of Tables
Preface to the second edition
Preface to the first edition
Commonly Used Notations
Chapter 1. An Introduction to the Homotopy Groups of Spheres
1. Classical Theorems Old and New
2. Methods of Computing
π
∗
(S
n
)
3. The Adams–Novikov
E
2
-term, Formal Group Laws, and the Greek
Letter Construction
4. More Formal Group Law Theory, Morava’s Point of View, and the
Chromatic Spectral Sequence
5. Unstable Homotopy Groups and the EHP Spectral Sequence
Chapter 2. Setting up the Adams Spectral Sequence
v
vii
ix
xi
xiii
1
2
5
12
20
24
41
1. The Classical Adams Spectral Sequence
41
Mod (p) Eilenberg–Mac Lane spectra. Mod (p) Adams resolutions. Differen-
tials. Homotopy inverse limits. Convergence. The extension problem. Examples:
integral and mod (p
i
) Eilenberg–Mac Lane spectra.
2. The Adams Spectral Sequence Based on a Generalized Homology
Theory
49
E
∗
-Adams resolutions.
E-completions.
The
E
∗
-Adams spectral sequence. As-
sumptions on the spectrum
E. E
∗
(E) is a Hopf algebroid. The canonical Adams
resolution. Convergence. The Adams filtration.
3. The Smash Product Pairing and the Generalized Connecting
Homomorphism
53
The smash product induces a pairing in the Adams spectral sequence. A map
that is trivial in homology raises Adams filtration. The connecting homomorphism
in Ext and the geometric boundary map.
Chapter 3. The Classical Adams Spectral Sequence
1. The Steenrod Algebra and Some Easy Calculations
2. The May Spectral Sequence
3. The Lambda Algebra
i
59
59
67
76
4. Some General Properties of Ext
5. Survey and Further Reading
Chapter 4.
BP
-Theory and the Adams–Novikov Spectral Sequence
84
92
101
1. Quillen’s Theorem and the Structure of
BP
∗
(BP )
101
Complex cobordism. Complex orientation of a ring spectrum. The formal
group law associated with a complex oriented homology theory. Quillen’s theorem
equating the Lazard and complex cobordism rings. Landweber and Novikov’s theo-
rem on the structure of
M U
∗
(M
U
). The Brown-Peterson spectrum
BP
. Quillen’s
idempotent operation and
p-typical
formal group laws. The structure of
BP
∗
(BP ).
2. A Survey of
BP
-Theory
109
Bordism groups of spaces. The Sullivan–Baas construction. The Johnson–
Wilson spectrum
BP n
. The Morava
K-theories K(n).
The Landweber filtration
and exact functor theorems. The Conner–Floyd isomorphism.
K-theory
as a func-
tor of complex cobordism. Johnson and Yosimura’s work on invariant regular ideals.
Infinite loop spaces associated with
M U
and
BP
; the Ravenel–Wilson Hopf ring.
The unstable Adams–Novikov spectral sequence of Bendersky, Curtis and Miller.
3. Some Calculations in
BP
∗
(BP )
115
The Morava-Landweber invariant prime ideal theorem. Some invariant regular
ideals. A generalization of Witt’s lemma. A formula for the universal
p-typical
formal group law. Formulas for the coproduct and conjugation in
BP
∗
(BP ). A
filtration of
BP
∗
(BP ))/I
n
.
4. Beginning Calculations with the Adams–Novikov Spectral Sequence
128
The Adams–Novikov spectral sequence and sparseness. The algebraic Novi-
kov spectral sequence of Novikov and Miller. Low dimensional Ext of the algebra
of Steenrod reduced powers. Bockstein spectral sequences leading to the Adams–
Novikov
E
2
-term. Calculations at odd primes. Toda’s theorem on the first non-
trivial odd primary Novikov differential. Chart for
p
= 5. Calculations and charts
for
p
= 2. Comparison with the Adams spectral sequence.
Chapter 5. The Chromatic Spectral Sequence
145
1. The Algebraic Construction
146
Greek letter elements and generalizations. The chromatic resolution, spectral
sequence, and cobar complex. The Morava stabilizer algebra Σ(n). The change-
of-rings theorem. The Morava vanishing theorem. Signs of Greek letter elements.
Computations with
β
t
. Decompsibility of
γ
1
. Chromatic differentials at
p
= 2.
Divisibility of
α
1
β
p
.
2. Ext
1
(BP
∗
/I
n
) and Hopf Invariant One
156
1
0
0
0
Ext (BP
∗
). Ext (M
1
). Ext (BP
∗
). Hopf invariant one elements. The Miller-
Wilson calculation of Ext
1
(BP
∗
/I
n
).
ii
3. Ext(M
1
) and the
J-Homomorphism
163
1
Ext(M ). Relation to im
J.
Patterns of differentials at
p
= 2. Computations
with the mod (2) Moore spectrum.
4. Ext
2
and the Thom Reduction
170
Results of Miller, Ravenel and Wilson (p
>
2) and Shimomura (p = 2) on
2
Ext (BP
∗
). Behavior of the Thom reduction map. Arf invariant differentials at
p >
2. Mahowald’s counterexample to the doomsday conjecture.
5. Periodic Families in Ext
2
175
Smith’s construction of
β
t
. Obstructions at
p
= 3. Results of Davis, Mahowald,
Oka, Smith and Zahler on permanent cycles in Ext
2
. Decomposables in Ext
2
.
6. Elements in Ext
3
and Beyond
181
3
4
Products of alphas and betas in Ext . Products of betas in Ext . A possible
obstruction to the existence of
V
(4).
Chapter 6. Morava Stabilizer Algebras
185
1. The Change-of-Rings Isomorphism
185
Theorems of Ravenel and Miller. Theorems of Morava. General nonsense about
Hopf algebroids. Formal group laws of Artin local rings. Morava’s proof. Miller
and Ravenel’s proof.
2. The Structure of Σ(n)
191
Relation to the group ring for
S
n
. Recovering the grading via an eigenspace de-
composition. A matrix representation of
S
n
. A splitting of
S
n
when
p| n.
Poincar´
/
e
duality and and periodic cohomology of
S
n
.
3. The Cohomology of Σ(n)
196
A May filtration of Σ(n) and the May spectral sequence. The open subgroup
theorem. Cohomology of some associated Lie algebras.
H
1
and
H
2
.
H
∗
(S(n)) for
n
= 1, 2, 3.
4. The Odd Primary Kervaire Invariant Elements
210
The nonexistence of certain elements and spectra. Detecting elements with the
cohomology of
Z/(p).
Differentials in the Adams spectral sequence.
5. The Spectra
T
(m)
218
A splitting theorem for certain Thom spectra. Application of the open subgroup
theorem. Ext
0
and Ext
1
.
Chapter 7. Computing Stable Homotopy Groups with the Adams–Novikov
Spectral Sequence
223
1.
2.
3.
4.
5.
6.
The method of infinite descent
2
The comodule
E
m+1
The homotopy of
T
(0)
(2)
and
T
(0)
(1)
The proof of Theorem 7.3.15
Computing
π
∗
(S
0
) for
p
= 3
Computations for
p
= 5
225
236
247
260
275
280
299
Appendix A1. Hopf Algebras and Hopf Algebroids
1. Basic Definitions
301
Hopf algebroids as cogroup objects in the category of commutative algebras.
Comodules. Cotensor products. Maps of Hopf algebroids. The associated Hopf
iii

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