nLab Modern foundations for stable homotopy theory

This pages collects links related to

• Anthony Elmendorf, Igor Kriz, Peter May,

  Modern foundations for stable homotopy theory,

  in Ioan Mackenzie James (ed.),

  Handbook of Algebraic Topology,

  1995

  Amsterdam: North-Holland, pp. 213–253,

  (pdf)

on the basics of stable homotopy theory in terms of the model category of S-modules with its symmetric smash product of spectra.

1. Spectra and the stable homotopy category

• generalized (Eilenberg-Steenrod) cohomology

• pointed object, loop space, suspension

• homotopy equivalence

• Brown representability theorem

• Omega spectrum, coordinate-free spectrum

• suspension spectrum

• homotopy classes

• homotopy group, stable homotopy groups of spheres

• weak homotopy equivalence

• stable homotopy category

• CW spectrum

• Whitehead theorem

2. Smash products and twisted half-smash products

• smash product

• smash product of spectra

• symmetric smash product of spectra

3. The category of $S$-modules and its derived category

• S-module, model structure on S-modules

4. The smash product of $S$-modules

• symmetric smash product of spectra

5. $A_\infty$ and $E_\infty$ ring spectra and their categories of modules

• A-∞ ring, E-∞ ring

• ∞-module

6. The smash product of $R$-modules and function $R$-modules

7. $Tor$ and $Ext$ in topology, and algebra.

• Tor, Ext

8.Universal coefficients and Künneth spectral sequence

• universal coefficient theorem

• Künneth spectral sequence

9. Algebraic constructions in the derived category of $R$-modules

• brave new algebra

10. Algebra structures on localizations and on quotients by ideals

11. The specialization of $MU$-modules and algebras

• MU

• Adams spectral sequence

• Adams-Novikov spectral sequence

category: reference