nLab Stable homotopy and generalised homology

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Contents

Context

Stable Homotopy theory

Cohomology

This page collects links related to

John Frank Adams,

Stable homotopy and generalized homology

Chicago Lectures in Mathematics, 1974

The University of Chicago Press 1974

ucp:bo21302708

pdf, pdf

on stable homotopy theory and generalised homology theory, with emphasis on complex cobordism theory, complex oriented cohomology theory, and the Adams spectral sequence/Adams-Novikov spectral sequence (today: “chromatic homotopy theory”).

Consists of three lectures, each meant to be readable on their own, and there is overlap in topics. It’s part III that begins with an actual introduction to stable homotopy theory, and so the beginner might prefer to start reading with Part III. Also notice that on p. 87 it says that the material there in part II is to be regarded as superseding part I.

A very detailed and readable account based on these lectures is

Stanley Kochmann, Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

The big story emerging here was later further developed in

Mike Hopkins, Complex oriented cohomology theories and the language of stacks, 1999
Doug Ravenel, Complex cobordism and stable homotopy groups of spheres, 1986/2003
Jacob Lurie, Chromatic Homotopy Theory, 2010

This is about understanding the absolute base space Spec(S) by covering it with Spec(MU). See at Adams spectral sequences – As derived descent.

Part I

2. Cobordism groups
3. Homology
4. The Conner-Floyd Chern classes
5. The Novikov operations
6. The algebra of all operations
7. Scholium on Novikov’s operations
8. Complex manifolds

Part II – Quillen’s work on formal groups and complex cobordism

0. Introduction
1. Formal groups
2. Examples from algebraic topology
3. Reformulation
4. Calculations in $E$ -homology and cohomology
5. Lazard’s universal ring
6. More calculations in $E$ -cohomology
7. The structure of Lazard’s universal ring $L$
8. Quillen’s theorem
9. Corollaries
10. Various formulae in $\pi_\bullet(MU)$
11. $MU_\bullet(MU)$
12. Behaviour of the Bott map
13. $K_\bullet(K)$
14. The Hattori-Stong theorem
15. Quillen’s idempotent cohomology operations
16. The Brown-Peterson spectrum
17. $KO_\bullet(KO)$

Part III

1. Introduction
2. Spectra
3. Elementary properties of the category of CW-spectra
4. Smash products
5. Spanier-Whitehead duality
6. Homology and cohomology
7. The Atiyah-Hirzebruch spectral sequence
8. The inverse limit and its derived functors
9. Products
10. Duality in manifolds
11. Applications in K-theory
12. The Steenrod algebra and its dual
13. A universal coefficient theorem
14. A category of fractions
15. The Adams spectral sequence
16. Applications to $\pi_\bullet(bu \wedge X)$ ; modules over $K[x,y]$
17. Structure of $\pi_\bullet(bu \wedge bu)$

Part I

2. Cobordism groups

3. Homology

Novikov operations?

4. The Conner-Floyd Chern classes

Conner-Floyd Chern class

5. The Novikov operations

6. The algebra of all operations

7. Scholium on Novikov’s operations

8. Complex manifolds

Umkehr map

Part II – Quillen’s work on formal groups and complex cobordism

0. Introduction

1. Formal groups

formal group law

2. Examples from algebraic topology

complex oriented cohomology theory
generalized Chern classes

3. Reformulation

4. Calculations in $E$ -homology and cohomology

universal coefficient theorem
universal complex orientation on MU (Lemma 4.6, example 4.7)

5. Lazard’s universal ring

Lazard ring

6. More calculations in $E$ -cohomology

7. The structure of Lazard’s universal ring $L$

Lazard's theorem

8. Quillen’s theorem

Quillen's theorem on MU

9. Corollaries

10. Various formulae in $\pi_\bullet(MU)$

11. $MU_\bullet(MU)$

Landweber-Novikov theorem

12. Behaviour of the Bott map

13. $K_\bullet(K)$

14. The Hattori-Stong theorem

15. Quillen’s idempotent cohomology operations

16. The Brown-Peterson spectrum

17. $KO_\bullet(KO)$

Part III

1. Introduction

smash product (of pointed topological spaces)

2. Spectra

3. Elementary properties of the category of CW-spectra

Adams category

There is much to love in his book, but not in the foundational part on CW spectra. (Peter May, MO comment)

4. Smash products

smash product of spectra

5. Spanier-Whitehead duality

Spanier-Whitehead duality

6. Homology and cohomology

7. The Atiyah-Hirzebruch spectral sequence

8. The inverse limit and its derived functors

derived functor, homotopy limit

9. Products

cup product

10. Duality in manifolds

11. Applications in K-theory

12. The Steenrod algebra and its dual

Steenrod algebra

13. A universal coefficient theorem

universal coefficient theorem

14. A category of fractions

What Adams tries to construct here – the localization of the stable homotopy category at the class of $E$ -equivalences – was later constructed by (Bousfield 79). See at Bousfield localization of spectra.

15. The Adams spectral sequence

Adams spectral sequence

16. Applications to $\pi_\bullet(bu \wedge X)$ ; modules over $K[x,y]$

17. Structure of $\pi_\bullet(bu \wedge bu)$

category: reference

Last revised on July 27, 2022 at 14:32:15. See the history of this page for a list of all contributions to it.

nLab Stable homotopy and generalised homology

Context

Stable Homotopy theory

Ingredients

Contents

Cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Part I

2. Cobordism groups

3. Homology

4. The Conner-Floyd Chern classes

5. The Novikov operations

6. The algebra of all operations

7. Scholium on Novikov’s operations

8. Complex manifolds

Part II – Quillen’s work on formal groups and complex cobordism

0. Introduction

1. Formal groups

2. Examples from algebraic topology

3. Reformulation

4. Calculations in EE-homology and cohomology

5. Lazard’s universal ring

6. More calculations in EE-cohomology

7. The structure of Lazard’s universal ring LL

8. Quillen’s theorem

9. Corollaries

10. Various formulae in π •(MU)\pi_\bullet(MU)

11. MU •(MU)MU_\bullet(MU)

12. Behaviour of the Bott map

13. K •(K)K_\bullet(K)

14. The Hattori-Stong theorem

15. Quillen’s idempotent cohomology operations

16. The Brown-Peterson spectrum

17. KO •(KO)KO_\bullet(KO)

Part III

1. Introduction

2. Spectra

3. Elementary properties of the category of CW-spectra

4. Smash products

5. Spanier-Whitehead duality

6. Homology and cohomology

7. The Atiyah-Hirzebruch spectral sequence

8. The inverse limit and its derived functors

9. Products

10. Duality in manifolds

11. Applications in K-theory

12. The Steenrod algebra and its dual

13. A universal coefficient theorem

14. A category of fractions

15. The Adams spectral sequence

16. Applications to π •(bu∧X)\pi_\bullet(bu \wedge X); modules over K[x,y]K[x,y]

17. Structure of π •(bu∧bu)\pi_\bullet(bu \wedge bu)

4. Calculations in $E$ -homology and cohomology

6. More calculations in $E$ -cohomology

7. The structure of Lazard’s universal ring $L$

10. Various formulae in $\pi_\bullet(MU)$

11. $MU_\bullet(MU)$

13. $K_\bullet(K)$

17. $KO_\bullet(KO)$

16. Applications to $\pi_\bullet(bu \wedge X)$ ; modules over $K[x,y]$

17. Structure of $\pi_\bullet(bu \wedge bu)$