nLab Complex cobordism and stable homotopy groups of spheres

Contents

Context

Homological algebra

Stable Homotopy theory

This entry collects pointers related to the book

Doug Ravenel

Complex cobordism and stable homotopy groups of spheres

Academic Press Orland (1986)

reprinted as:

AMS Chelsea Publishing, Volume 347 (2004)

ISBN:978-0-8218-2967-7

(webpage, pdf)

on stable homotopy theory in general and in particular the computation of the homotopy groups of spheres via the Adams-Novikov spectral sequence and its use of complex cobordism cohomology theory.

My initial inclination was to call this book The Music of the Spheres, but I was dissuaded from doing so by my diligent publisher, who is ever mindful of the sensibilities of librarians. (preface to the first edition)

Chapter 1. An introduction to the Homotopy Groups of Spheres

1. Classical theorems Old and New

2. Methods of computing $\pi_\ast(S^n)$

3. The Adams-Novikov $E_2$ -term, Formal Group Laws, and the Greek Letter Construction

formal group law
Lazard ring
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

EHP spectral sequence

Chapter 2. Setting up the Adams Spectral sequence

1. The classical Adams spectral sequence

2. The Adams spectral sequence based on a generalized homology theory

commutative Hopf algebroid structure on Steenrod algebra

3. The smash product pairing and the Generalized connecting homomorphism

connecting homomorphism

Chapter 3. The Classical Adams Spectral Sequence

classical Adams spectral sequence

1. The Steenrod algebra and some easy calculuation

Hopf algebra structure on Steenrod algebra
cobar complex?, cosimplicial object

2. The May spectral sequence

3. The Lambda Algebra

4. Some general properties of $Ext$

periodicity theorem

5. Survey and further reading

Chapter 4. $B P$ -Theory and the Adams-Novikov Spectral Sequence

Adams-Novikov spectral sequence

1. Quillen’s theorem and the structure of $BP_\bullet(BP)$

2. A survey of $B P$ -theory

3. Some calculations in $B P_\bullet(B P)$

4. Beginning calculations with the Adams-Novikov Spectral Sequence

BP-Adams-Novikov spectral sequence

Chapter 5. The Chromatic Spectral Sequence

4. $Ext^2$ and the Thom Reduction

5. Periodic families in $Ext^2$

6. Elements in $Ext^3$ and Beyond

Chapter 6. Morava Stabilizer Algebras

Morava stabilizer group

1. The Change-of-Rings Isomorphism

2. The Structure of $\Sigma(n)$

3. The Cohomology of $\Sigma(n)$

4. The Odd Primary Kervaire Invariant Elements

5. The Spectra $T(m)$

Ravenel's spectra

Chapter 7. Computing Stable Homotopy Groups with the Adams-Novikov Spectral Sequence

Appendix 1. Hopf Algebras and Hopf Algebroids

1. Basic definitions

2. Homological algebra

derived functor
Ext and Cotor for comodules

3. Some spectral sequences

4. Massey products

Massey product

5. Algebraic Steenrod operations

Steenrod algebra on Cotor-groups of comodules over commutative Hopf algebroids

Appendix 2. Formal Group Laws

formal group law

Appendix 3. Table of homotopy groups of spheres

category: reference

Last revised on December 30, 2020 at 08:36:40. See the history of this page for a list of all contributions to it.

nLab Complex cobordism and stable homotopy groups of spheres

Context

Homological algebra

Stable Homotopy theory

Ingredients

Contents

Contents

Chapter 1. An introduction to the Homotopy Groups of Spheres

1. Classical theorems Old and New

2. Methods of computing π *(S n)\pi_\ast(S^n)

3. The Adams-Novikov E 2E_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

1. The classical Adams spectral sequence

2. The Adams spectral sequence based on a generalized homology theory

3. The smash product pairing and the Generalized connecting homomorphism

Chapter 3. The Classical Adams Spectral Sequence

1. The Steenrod algebra and some easy calculuation

2. The May spectral sequence

3. The Lambda Algebra

4. Some general properties of ExtExt

5. Survey and further reading

Chapter 4. BPB P-Theory and the Adams-Novikov Spectral Sequence

1. Quillen’s theorem and the structure of BP •(BP)BP_\bullet(BP)

2. A survey of BPB P-theory

3. Some calculations in BP •(BP)B P_\bullet(B P)

4. Beginning calculations with the Adams-Novikov Spectral Sequence

Chapter 5. The Chromatic Spectral Sequence

1. The algebraic construction

2. Ext 1(BP •/I n)Ext^1(B P_\bullet/I_n) and Hopf Invariant One

3. Ext(M 1)Ext(M^1) and the JJ-Homomorphism

4. Ext 2Ext^2 and the Thom Reduction

5. Periodic families in Ext 2Ext^2

6. Elements in Ext 3Ext^3 and Beyond

Chapter 6. Morava Stabilizer Algebras

1. The Change-of-Rings Isomorphism

2. The Structure of Σ(n)\Sigma(n)

3. The Cohomology of Σ(n)\Sigma(n)

4. The Odd Primary Kervaire Invariant Elements

5. The Spectra T(m)T(m)

Chapter 7. Computing Stable Homotopy Groups with the Adams-Novikov Spectral Sequence

Appendix 1. Hopf Algebras and Hopf Algebroids

1. Basic definitions

2. Homological algebra

3. Some spectral sequences

4. Massey products

5. Algebraic Steenrod operations

Appendix 2. Formal Group Laws

Appendix 3. Table of homotopy groups of spheres

2. Methods of computing $\pi_\ast(S^n)$

3. The Adams-Novikov $E_2$ -term, Formal Group Laws, and the Greek Letter Construction

4. Some general properties of $Ext$

Chapter 4. $B P$ -Theory and the Adams-Novikov Spectral Sequence

1. Quillen’s theorem and the structure of $BP_\bullet(BP)$

2. A survey of $B P$ -theory

3. Some calculations in $B P_\bullet(B P)$

2. $Ext^1(B P_\bullet/I_n)$ and Hopf Invariant One

3. $Ext(M^1)$ and the $J$ -Homomorphism

4. $Ext^2$ and the Thom Reduction

5. Periodic families in $Ext^2$

6. Elements in $Ext^3$ and Beyond

2. The Structure of $\Sigma(n)$

3. The Cohomology of $\Sigma(n)$

5. The Spectra $T(m)$