nLab Stable homotopy and generalised homology



Stable Homotopy theory



Special and general types

Special notions


Extra structure



This page collects links related to

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

The big story emerging here was later further developed in

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

  • Novikov operations?

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

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

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

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

15. The Adams spectral sequence

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

17. Structure of π (bubu)\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.