Complex cobordism and stable homotopy groups of spheres:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
AMS Chelsea Publ.
2004
|
| Edition: | 2. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XIX, 395 S. graph. Darst. |
| ISBN: | 082182967X |
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| 100 | 1 | |a Ravenel, Douglas C. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Complex cobordism and stable homotopy groups of spheres |c Douglas C. Ravenel |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Providence, RI |b AMS Chelsea Publ. |c 2004 | |
| 300 | |a XIX, 395 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Cobordism theory | |
| 650 | 4 | |a Homotopy groups | |
| 650 | 4 | |a Spectral sequences (Mathematics) | |
| 650 | 4 | |a Sphere | |
| 650 | 0 | 7 | |a Spektralsequenz |0 (DE-588)4182172-5 |2 gnd |9 rswk-swf |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-015016121 | ||
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| adam_text | Contents
List of Figures
xi
List of Tables
xiii
Preface to the Second Edition
xv
Preface to the First Edition
xvii
Commonly Used Notations
xix
Chapter
1.
An Introduction to the Homotopy Groups of Spheres
1
1.
Classical Theorems Old and New
2
Homotopy groups. The Hurewicz and
Freudenthal
theorems. Stable
stems. The
Hopf map.
Serre s finiteness theorem. Nishida s nilpotence
theorem. Cohen, Moore and Neisendorfer s exponent theorem.
Bott
per¬
iodicity. The J-homomorphism.
2.
Methods of Computing
π»
(Sn)
5
Eilenberg-Mac Lane spaces and Serre s method. The Adams spectral
sequence.
Hopf
invariant one theorems. The Adams—Novikov spectral
sequence. Tables in low dimensions for
ρ
= 3.
3.
The Adams-Novikov
Ą-term,
Formal Group Laws,
and the Greek Letter Construction
12
Formal group laws and Quillen s theorem. The Adams-Novikov E2-
term as group cohomology. Alphas, betas and gammas. The
Morava-
Landweber
theorem and higher Greek letters. Generalized Greek letter
elements.
4.
More Formal Group Law Theory, Morava s Point of View, and the
Chromatic Spectral Sequence
19
The Brown-Peterson spectrum. Classification of formal group laws.
Morava s group action, its orbits and stabilizers. The chromatic resolution
and the chromatic spectral sequence.
Bockstein
spectral sequences. Use
of cyclic subgroups to detect Arf invariant elements. Morava s vanishing
theorem. Greek letter elements in the chromatic spectral sequence.
5.
Unstable Homotopy Groups and the EHP Spectral Sequence
24
The EHP sequences. The EHP spectral sequence. The stable zone.
The inductive method. The stable EHP spectral sequence. The Adams
vector field theorem. James periodicity. The J-spectrum. The spectral
vi
CONTENTS
sequence for ^(RP00) and J«(5EP). Relation to the Segal conjecture.
The Mahowald root invariant.
Chapter
2.
Setting up the Adams Spectral Sequence
41
1.
The Classical Adams Spectral Sequence
41
Mod (p) Eilenberg-Mac Lane spectra. Mod (p) Adams resolutions.
Differentials. Homotopy inverse limits. Convergence. The extension prob¬
lem. Examples: integral and mod
(рг)
Eilenberg-Mac Lane spectra.
2.
The Adams Spectral Sequence Based on a Generalized Homology
Theory
49
jE^-Adams resolutions. E-completions. The JS^-Adams spectral se¬
quence. Assumptions 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
59
1.
The Steenrod Algebra and Some Easy Calculations
59
Milnor s structure theorem for A*. The cobar complex. Multiplica¬
tion by
ρ
in the
.Eoo
-term. The Adams spectral sequence for
ћџ(М1Ј).
Computations for MO, bu and bo.
2.
The May Spectral Sequence
67
May s filtration of A». Nonassociativity of May s Ei-term and a way
to avoid it. Computations at
ρ
= 2
in low dimensions. Computations with
the
subalgebra
A(2) at
ρ
— 2.
3.
The Lambda Algebra
74
Λ
as an Adams Si-term. The algebraic EHP spectral sequence.
Serial numbers. The Curtis algorithm. Computations below dimension
14.
James periodicity. The Adams vanishing line. d is multiplication by
λ_ι.
Illustration for Sz.
4.
Some General Properties of Ext
85
Exts for
s
< 3.
Behavior of elements in Ext2. Adams vanishing line
of slope
1/2
for
ρ
= 2.
Periodicity above a line of slope
1/5
for
ρ
= 2.
Elements not annihilated by any periodicity operators and their relation
to
im J.
An elementary proof that most of these elements are
nontrivial.
5.
Survey and Further Reading
94
Exotic cobordism theories. Decreasing nitrations of A» and the result¬
ing spectral sequences. Application to MSp.
Mahowalďs
generalizations
CONTENTS
vii
of
Λ.
^„-periodicity in the Adams spectral sequence. Selected references
to related work.
Chapter
4.
BP-Theory and the Adams-Novikov Spectral Sequence
103
1.
Quillen s Theorem and the Structure of
BPĄBP)
103
Complex cobordism. Complex orientation of a ring spectrum. The
formal group law for a complex oriented homology theory. Quillen s the¬
orem equating the
Lazard
and complex cobordism rings.
Landweber
and
Novikov s theorem on the structure of
MU^(MU).
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
111
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 iso¬
morphism,
.ří-theory
as a functor of complex cobordism. Johnson and
Yosimura s work on invariant regular ideals. Infinite loop spaces associ¬
ated with MU and BP; the
Ravenel-Wilson Hopf
ring. The unstable
Adams-Novikov spectral sequence of Bendersky, Curtis and Miller.
3.
Some Calculations in BP*(BP)
117
The Morava-Landweber invariant prime ideal theorem. Some invari¬
ant regular ideals. A generalization of Witt s lemma. A formula for the
universal p-typical formal group law. Formulas for the coproduct and con¬
jugation in
ВР„(ВР).
A filtration of BP*(BP)/In.
4.
Beginning Calculations with the Adams-Novikov Spectral Sequence
130
The Adams-Novikov spectral sequence and sparseness. The alge¬
braic Novikov spectral sequence of Novikov and Miller. Low dimensional
Ext of the algebra of Steenrod reduced powers.
Bockstein
spectral se¬
quences leading to the Adams-Novikov i^-term. Calculations at odd
primes. Toda s theorem on the first
nontrivial
odd primary Novikov dif¬
ferential. Chart for
ρ
= 5.
Calculations and charts for
ρ
= 2.
Comparison
with the Adams spectral sequence.
Chapter
5.
The Chromatic Spectral Sequence
147
1.
The Algebraic Construction
148
Greek letter elements and generalizations. The chromatic resolu¬
tion, spectral sequence, and cobar complex. The
Morava
stabilizer alge¬
bra
Σ(η).
The change-of-rings theorem. The
Morava
vanishing theorem.
Signs of Greek letter elements. Computations with
ßt. Decomposability
of 7i. Chromatic differentials at
ρ
= 2.
Divisibility of
α. βν.
2.
Ext1 (BP*/In) and
Hopf
Invariant One
158
Ext°(SP*).
Ext^Mf).
Ext1
(SĄ).
Hopf
invariant one elements.
The Miller-Wilson calculation of Ext1 (BP*
/
In)
.
viii CONTENTS
3. Ext(M:)
and the J-Homomorphism
165
Ext(Mx). Relation to
im J.
Patterns of differentials at
p
= 2.
Comp¬
utations with the mod
(2)
Moore spectrum.
4.
Ext2 and the Thorn Reduction
172
Results of Miller, Ravenel and Wilson (p
> 2)
and Shimomura (p
= 2)
on Ext2(SP*). Behavior of the Thorn reduction map. Arf invariant differ¬
entials at
p
> 2.
Mahowald s counterexample to the doomsday conjecture.
5.
Periodic Families in Ext2
178
Smith s construction of
ßt.
Obstructions at
p
= 3.
Results of Davis,
Mahowald, Oka, Smith and
Zahler
on permanent cycles in Ext2. Decom-
posables in Ext2.
6.
Elements in Ext3 and Beyond
183
Products of alphas and betas in Ext3. Products of betas in Ext4. A
possible obstruction to the existence of V(4).
Chapter
6.
Morava
Stabilizer Algebras
187
1.
The Change-of-Rings Isomorphism
188
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{η)
192
Relation to the group ring for 5n. Recovering the grading via an
eigenspace decomposition. A matrix representation of Sn. A splitting of
Sn when
p n. Poincaré
duality and periodic cohomology of Sn.
3.
The Cohomology of
Σ(η)
198
A May filtration of
Σ(η)
and the May spectral sequence. The open
subgroup theorem. Cohomology of some associated Lie algebras.
Я1
and
ff2. H*{S{n))
forra
= 1,2,3.
4.
The Odd Primary Kervaire Invariant Elements
212
The nonexistence of certain elements and spectra. Detecting ele¬
ments with the cohomology of Z/(p). Differentials in the Adams spectral
sequence.
5.
The Spectra T{m)
219
A splitting theorem for certain Thorn spectra. Application of the
open subgroup theorem. Ext0 and Ext1
.
Chapter
7.
Computing Stable Homotopy Groups with the Adams-Novikov
Spectral Sequence
225
1.
The method of infinite descent
227
r(m
+ 1),
A(m), and G(m
+
l,k
— 1).
Weak injective comodules.
Quillen operations,
ż-free
comodules. The small descent spectral sequence
CONTENTS ix
and topological small descent spectral sequence. Input/output procedure.
The 4-term exact sequence. Hat notation. A generalization of the Morava-
Landweber theorem. The
A¿
and
Ο^+1.
Poincaré
series. ExtjL +1).
Properties of weak injectives.
2.
The comodule E^+1
238
D . The
Poincaré
series for E^+1.
Ext(£ľ^+1)
below dimension
p2|t?i |. The comodules
В
and U.
3.
The homotopy of T(0)(2) and T{0)(1)
249
The j-freeness of B. The elements Uij. A short exact sequence for
U.
7г*(Г(0)(2)).
Cartan-Eilenberg differentials for
Г(0)(і).
4.
The proof of Theorem
7.3.15 261
P(l)*.
£?,
Ď ,
and
£?.
P-free comodules. A filtration of
Ď .
A
4-term exact sequence of P(l)*-comodules.
C¿
and the skeletal filtration
spectral sequence.
5.
Computing
тг*
(S°) for
ρ
= 3 277
The input list I. Computation of differentials in the topological small
descent spectral sequence.
6.
Computations for
ρ
= 5 282
The differential on
73
and the nonexistence of V(3) for
ρ
= 5.
The
input list I. Differentials.
Appendix
AI. Hopf
Algebras and
Hopf Algebroids 299
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
algebra. Normal maps. Unicursal
Hopf
algebroids. The
kernel of a normal map.
Hopf algebroid
extensions. The comodule algebra
structure theorem. Invariant ideals. Split
Hopf
algebroids.
2.
Homological Algebra
309
Injective comodules. The derived functors
Cotor
and Ext. Rela¬
tive injectives and resolutions. The cobar resolution and complex. Cup
products. Ext isomorphisms for invariant ideals and split
Hopf
algebroids.
3.
Some Spectral Sequences
315
The resolution spectral sequence. Filtered
Hopf
algebroids. Filtra-
tions by powers of the unit coideal. The spectral sequence associated
with
a
Hopf
algebroid map. Change-of-rings isomorphism. The Cartan-
Eilenberg spectral sequence. A formulation due to Adams. The
Ą-term
for a cocentral extension of
Hopf
algebras.
4.
Massey Products
323
Definitions of
η
-fold Massey products and indeterminacy. Defining
systems. Juggling theorems: associativity and commutativity formulas.
χ
CONTENTS
Convergence of Massey products in spectral sequences. A Leibnitz formula
for differentials. Differentials and extensions in spectral sequences.
5.
Algebraic Steenrod Operations
332
Construction, Cartan formula and
Adem
relations. Commutativity
with suspension. Kudo transgression theorem.
Appendix A2. Formal Group Laws
339
1.
Universal Formal Group Laws and Strict Isomorphisms
339
Definition and examples of formal group laws. Homomorphisms,
isomorphisms and logarithms. The universal formal group law and the
Lazard
ring. Lazard s comparison lemma. The
Hopf
algebroid VT. Proof
of the comparison lemma.
2.
Classification and Endomorphism Rings
351
Hazewinkel s and Araki s generators. The right unit formula. The
height of a formal group law. Classification in characteristic p. Finite
fields, Witt rings and division algebras. The endomorphism ring of a
height
η
formal group law.
Appendix
A3.
Tables of Homotopy Groups of Spheres
361
The Adams spectral sequence for
ρ
= 2
below dimension
62.
The
Adams-Novikov spectral sequence for
ρ
= 2
below dimension
40.
Comp¬
arison of
Toda s, Tangora s
and our notation at
ρ
= 2.
3-Primary stable
homotopy excluding
im J.
5-Primary stable homotopy excluding
im J.
Bibliography
377
Index
391
|
| adam_txt |
Contents
List of Figures
xi
List of Tables
xiii
Preface to the Second Edition
xv
Preface to the First Edition
xvii
Commonly Used Notations
xix
Chapter
1.
An Introduction to the Homotopy Groups of Spheres
1
1.
Classical Theorems Old and New
2
Homotopy groups. The Hurewicz and
Freudenthal
theorems. Stable
stems. The
Hopf map.
Serre's finiteness theorem. Nishida's nilpotence
theorem. Cohen, Moore and Neisendorfer's exponent theorem.
Bott
per¬
iodicity. The J-homomorphism.
2.
Methods of Computing
π»
(Sn)
5
Eilenberg-Mac Lane spaces and Serre's method. The Adams spectral
sequence.
Hopf
invariant one theorems. The Adams—Novikov spectral
sequence. Tables in low dimensions for
ρ
= 3.
3.
The Adams-Novikov
Ą-term,
Formal Group Laws,
and the Greek Letter Construction
12
Formal group laws and Quillen's theorem. The Adams-Novikov E2-
term as group cohomology. Alphas, betas and gammas. The
Morava-
Landweber
theorem and higher Greek letters. Generalized Greek letter
elements.
4.
More Formal Group Law Theory, Morava's Point of View, and the
Chromatic Spectral Sequence
19
The Brown-Peterson spectrum. Classification of formal group laws.
Morava's group action, its orbits and stabilizers. The chromatic resolution
and the chromatic spectral sequence.
Bockstein
spectral sequences. Use
of cyclic subgroups to detect Arf invariant elements. Morava's vanishing
theorem. Greek letter elements in the chromatic spectral sequence.
5.
Unstable Homotopy Groups and the EHP Spectral Sequence
24
The EHP sequences. The EHP spectral sequence. The stable zone.
The inductive method. The stable EHP spectral sequence. The Adams
vector field theorem. James periodicity. The J-spectrum. The spectral
vi
CONTENTS
sequence for ^(RP00) and J«(5EP). Relation to the Segal conjecture.
The Mahowald root invariant.
Chapter
2.
Setting up the Adams Spectral Sequence
41
1.
The Classical Adams Spectral Sequence
41
Mod (p) Eilenberg-Mac Lane spectra. Mod (p) Adams resolutions.
Differentials. Homotopy inverse limits. Convergence. The extension prob¬
lem. Examples: integral and mod
(рг)
Eilenberg-Mac Lane spectra.
2.
The Adams Spectral Sequence Based on a Generalized Homology
Theory
49
jE^-Adams resolutions. E-completions. The JS^-Adams spectral se¬
quence. Assumptions 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
59
1.
The Steenrod Algebra and Some Easy Calculations
59
Milnor's structure theorem for A*. The cobar complex. Multiplica¬
tion by
ρ
in the
.Eoo
-term. The Adams spectral sequence for
ћџ(М1Ј).
Computations for MO, bu and bo.
2.
The May Spectral Sequence
67
May's filtration of A». Nonassociativity of May's Ei-term and a way
to avoid it. Computations at
ρ
= 2
in low dimensions. Computations with
the
subalgebra
A(2) at
ρ
— 2.
3.
The Lambda Algebra
74
Λ
as an Adams Si-term. The algebraic EHP spectral sequence.
Serial numbers. The Curtis algorithm. Computations below dimension
14.
James periodicity. The Adams vanishing line. d\ is multiplication by
λ_ι.
Illustration for Sz.
4.
Some General Properties of Ext
85
Exts for
s
< 3.
Behavior of elements in Ext2. Adams' vanishing line
of slope
1/2
for
ρ
= 2.
Periodicity above a line of slope
1/5
for
ρ
= 2.
Elements not annihilated by any periodicity operators and their relation
to
im J.
An elementary proof that most of these elements are
nontrivial.
5.
Survey and Further Reading
94
Exotic cobordism theories. Decreasing nitrations of A» and the result¬
ing spectral sequences. Application to MSp.
Mahowalďs
generalizations
CONTENTS
vii
of
Λ.
^„-periodicity in the Adams spectral sequence. Selected references
to related work.
Chapter
4.
BP-Theory and the Adams-Novikov Spectral Sequence
103
1.
Quillen's Theorem and the Structure of
BPĄBP)
103
Complex cobordism. Complex orientation of a ring spectrum. The
formal group law for a complex oriented homology theory. Quillen's the¬
orem equating the
Lazard
and complex cobordism rings.
Landweber
and
Novikov's theorem on the structure of
MU^(MU).
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
111
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 iso¬
morphism,
.ří-theory
as a functor of complex cobordism. Johnson and
Yosimura's work on invariant regular ideals. Infinite loop spaces associ¬
ated with MU and BP; the
Ravenel-Wilson Hopf
ring. The unstable
Adams-Novikov spectral sequence of Bendersky, Curtis and Miller.
3.
Some Calculations in BP*(BP)
117
The Morava-Landweber invariant prime ideal theorem. Some invari¬
ant regular ideals. A generalization of Witt's lemma. A formula for the
universal p-typical formal group law. Formulas for the coproduct and con¬
jugation in
ВР„(ВР).
A filtration of BP*(BP)/In.
4.
Beginning Calculations with the Adams-Novikov Spectral Sequence
130
The Adams-Novikov spectral sequence and sparseness. The alge¬
braic Novikov spectral sequence of Novikov and Miller. Low dimensional
Ext of the algebra of Steenrod reduced powers.
Bockstein
spectral se¬
quences leading to the Adams-Novikov i^-term. Calculations at odd
primes. Toda's theorem on the first
nontrivial
odd primary Novikov dif¬
ferential. Chart for
ρ
= 5.
Calculations and charts for
ρ
= 2.
Comparison
with the Adams spectral sequence.
Chapter
5.
The Chromatic Spectral Sequence
147
1.
The Algebraic Construction
148
Greek letter elements and generalizations. The chromatic resolu¬
tion, spectral sequence, and cobar complex. The
Morava
stabilizer alge¬
bra
Σ(η).
The change-of-rings theorem. The
Morava
vanishing theorem.
Signs of Greek letter elements. Computations with
ßt. Decomposability
of 7i. Chromatic differentials at
ρ
= 2.
Divisibility of
α.\βν.
2.
Ext1 (BP*/In) and
Hopf
Invariant One
158
Ext°(SP*).
Ext^Mf).
Ext1
(SĄ).
Hopf
invariant one elements.
The Miller-Wilson calculation of Ext1 (BP*
/
In)
.
viii CONTENTS
3. Ext(M:)
and the J-Homomorphism
165
Ext(Mx). Relation to
im J.
Patterns of differentials at
p
= 2.
Comp¬
utations with the mod
(2)
Moore spectrum.
4.
Ext2 and the Thorn Reduction
172
Results of Miller, Ravenel and Wilson (p
> 2)
and Shimomura (p
= 2)
on Ext2(SP*). Behavior of the Thorn reduction map. Arf invariant differ¬
entials at
p
> 2.
Mahowald's counterexample to the doomsday conjecture.
5.
Periodic Families in Ext2
178
Smith's construction of
ßt.
Obstructions at
p
= 3.
Results of Davis,
Mahowald, Oka, Smith and
Zahler
on permanent cycles in Ext2. Decom-
posables in Ext2.
6.
Elements in Ext3 and Beyond
183
Products of alphas and betas in Ext3. Products of betas in Ext4. A
possible obstruction to the existence of V(4).
Chapter
6.
Morava
Stabilizer Algebras
187
1.
The Change-of-Rings Isomorphism
188
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{η)
192
Relation to the group ring for 5n. Recovering the grading via an
eigenspace decomposition. A matrix representation of Sn. A splitting of
Sn when
p \ n. Poincaré
duality and periodic cohomology of Sn.
3.
The Cohomology of
Σ(η)
198
A May filtration of
Σ(η)
and the May spectral sequence. The open
subgroup theorem. Cohomology of some associated Lie algebras.
Я1
and
ff2. H*{S{n))
forra
= 1,2,3.
4.
The Odd Primary Kervaire Invariant Elements
212
The nonexistence of certain elements and spectra. Detecting ele¬
ments with the cohomology of Z/(p). Differentials in the Adams spectral
sequence.
5.
The Spectra T{m)
219
A splitting theorem for certain Thorn spectra. Application of the
open subgroup theorem. Ext0 and Ext1
.
Chapter
7.
Computing Stable Homotopy Groups with the Adams-Novikov
Spectral Sequence
225
1.
The method of infinite descent
227
r(m
+ 1),
A(m), and G(m
+
l,k
— 1).
Weak injective comodules.
Quillen operations,
ż-free
comodules. The small descent spectral sequence
CONTENTS ix
and topological small descent spectral sequence. Input/output procedure.
The 4-term exact sequence. Hat notation. A generalization of the Morava-
Landweber theorem. The
A¿
and
Ο^+1.
Poincaré
series. ExtjL +1).
Properties of weak injectives.
2.
The comodule E^+1
238
D\. The
Poincaré
series for E^+1.
Ext(£ľ^+1)
below dimension
p2|t?i |. The comodules
В
and U.
3.
The homotopy of T(0)(2) and T{0)(1)
249
The j-freeness of B. The elements Uij. A short exact sequence for
U.
7г*(Г(0)(2)).
Cartan-Eilenberg differentials for
Г(0)(і).
4.
The proof of Theorem
7.3.15 261
P(l)*.
£?,
Ď\,
and
£?.
P-free comodules. A filtration of
Ď\.
A
4-term exact sequence of P(l)*-comodules.
C¿
and the skeletal filtration
spectral sequence.
5.
Computing
тг*
(S°) for
ρ
= 3 277
The input list I. Computation of differentials in the topological small
descent spectral sequence.
6.
Computations for
ρ
= 5 282
The differential on
73
and the nonexistence of V(3) for
ρ
= 5.
The
input list I. Differentials.
Appendix
AI. Hopf
Algebras and
Hopf Algebroids 299
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
algebra. Normal maps. Unicursal
Hopf
algebroids. The
kernel of a normal map.
Hopf algebroid
extensions. The comodule algebra
structure theorem. Invariant ideals. Split
Hopf
algebroids.
2.
Homological Algebra
309
Injective comodules. The derived functors
Cotor
and Ext. Rela¬
tive injectives and resolutions. The cobar resolution and complex. Cup
products. Ext isomorphisms for invariant ideals and split
Hopf
algebroids.
3.
Some Spectral Sequences
315
The resolution spectral sequence. Filtered
Hopf
algebroids. Filtra-
tions by powers of the unit coideal. The spectral sequence associated
with
a
Hopf
algebroid map. Change-of-rings isomorphism. The Cartan-
Eilenberg spectral sequence. A formulation due to Adams. The
Ą-term
for a cocentral extension of
Hopf
algebras.
4.
Massey Products
323
Definitions of
η
-fold Massey products and indeterminacy. Defining
systems. Juggling theorems: associativity and commutativity formulas.
χ
CONTENTS
Convergence of Massey products in spectral sequences. A Leibnitz formula
for differentials. Differentials and extensions in spectral sequences.
5.
Algebraic Steenrod Operations
332
Construction, Cartan formula and
Adem
relations. Commutativity
with suspension. Kudo transgression theorem.
Appendix A2. Formal Group Laws
339
1.
Universal Formal Group Laws and Strict Isomorphisms
339
Definition and examples of formal group laws. Homomorphisms,
isomorphisms and logarithms. The universal formal group law and the
Lazard
ring. Lazard's comparison lemma. The
Hopf
algebroid VT. Proof
of the comparison lemma.
2.
Classification and Endomorphism Rings
351
Hazewinkel's and Araki's generators. The right unit formula. The
height of a formal group law. Classification in characteristic p. Finite
fields, Witt rings and division algebras. The endomorphism ring of a
height
η
formal group law.
Appendix
A3.
Tables of Homotopy Groups of Spheres
361
The Adams spectral sequence for
ρ
= 2
below dimension
62.
The
Adams-Novikov spectral sequence for
ρ
= 2
below dimension
40.
Comp¬
arison of
Toda's, Tangora's
and our notation at
ρ
= 2.
3-Primary stable
homotopy excluding
im J.
5-Primary stable homotopy excluding
im J.
Bibliography
377
Index
391 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Ravenel, Douglas C. |
| author_facet | Ravenel, Douglas C. |
| author_role | aut |
| author_sort | Ravenel, Douglas C. |
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| building | Verbundindex |
| bvnumber | BV021803685 |
| callnumber-first | Q - Science |
| callnumber-label | QA612 |
| callnumber-raw | QA612.78 |
| callnumber-search | QA612.78 |
| callnumber-sort | QA 3612.78 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 300 |
| classification_tum | MAT 553f |
| ctrlnum | (OCoLC)53145258 (DE-599)BVBBV021803685 |
| dewey-full | 514/.24 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 514 - Topology |
| dewey-raw | 514/.24 |
| dewey-search | 514/.24 |
| dewey-sort | 3514 224 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV021803685 |
| illustrated | Illustrated |
| index_date | 2024-07-02T15:48:42Z |
| indexdate | 2024-07-09T20:44:59Z |
| institution | BVB |
| isbn | 082182967X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015016121 |
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| spelling | Ravenel, Douglas C. Verfasser aut Complex cobordism and stable homotopy groups of spheres Douglas C. Ravenel 2. ed. Providence, RI AMS Chelsea Publ. 2004 XIX, 395 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Cobordism theory Homotopy groups Spectral sequences (Mathematics) Sphere Spektralsequenz (DE-588)4182172-5 gnd rswk-swf Stabile Homotopiegruppe (DE-588)4234506-6 gnd rswk-swf Kugel (DE-588)4165914-4 gnd rswk-swf Homotopiegruppe (DE-588)4160621-8 gnd rswk-swf Stabile Homotopiegruppe (DE-588)4234506-6 s Kugel (DE-588)4165914-4 s DE-604 Homotopiegruppe (DE-588)4160621-8 s Spektralsequenz (DE-588)4182172-5 s Erscheint auch als Online-Ausgabe 978-1-4704-2998-0 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015016121&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ravenel, Douglas C. Complex cobordism and stable homotopy groups of spheres Cobordism theory Homotopy groups Spectral sequences (Mathematics) Sphere Spektralsequenz (DE-588)4182172-5 gnd Stabile Homotopiegruppe (DE-588)4234506-6 gnd Kugel (DE-588)4165914-4 gnd Homotopiegruppe (DE-588)4160621-8 gnd |
| subject_GND | (DE-588)4182172-5 (DE-588)4234506-6 (DE-588)4165914-4 (DE-588)4160621-8 |
| title | Complex cobordism and stable homotopy groups of spheres |
| title_auth | Complex cobordism and stable homotopy groups of spheres |
| title_exact_search | Complex cobordism and stable homotopy groups of spheres |
| title_exact_search_txtP | Complex cobordism and stable homotopy groups of spheres |
| title_full | Complex cobordism and stable homotopy groups of spheres Douglas C. Ravenel |
| title_fullStr | Complex cobordism and stable homotopy groups of spheres Douglas C. Ravenel |
| title_full_unstemmed | Complex cobordism and stable homotopy groups of spheres Douglas C. Ravenel |
| title_short | Complex cobordism and stable homotopy groups of spheres |
| title_sort | complex cobordism and stable homotopy groups of spheres |
| topic | Cobordism theory Homotopy groups Spectral sequences (Mathematics) Sphere Spektralsequenz (DE-588)4182172-5 gnd Stabile Homotopiegruppe (DE-588)4234506-6 gnd Kugel (DE-588)4165914-4 gnd Homotopiegruppe (DE-588)4160621-8 gnd |
| topic_facet | Cobordism theory Homotopy groups Spectral sequences (Mathematics) Sphere Spektralsequenz Stabile Homotopiegruppe Kugel Homotopiegruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015016121&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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