nLab
Adams–Novikov spectral sequence

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Stable Homotopy theory

Contents

Idea

The Adams-Novikov spectral sequence is a class of spectral sequences which converge to and hence are used to compute homotopy groups of connective spectra, hence in particular the stable homotopy groups of spheres. It refines the Adams spectral sequence by replacing ordinary cohomology with coefficients in /p\mathbb{Z}/p\mathbb{Z} bycomplex cobordism cohomology theory.

More generally for EE any suitable E-infinity ring there is an Adams-Novikov-type spectral sequence involving EE-generalized cohomology/generalized homology. This fully general notion is often again just referred to as the EE-Adams spectral sequence. Accordingly, see there for more.

Properties

Relation to Brown-Peterson spectrum

The pp-component of the E 2E^2-term of the Adams-Novikov spectral sequence for the sphere spectrum, hence the one converging to the stable homotopy groups of spheres π *(𝕊)\pi_\ast(\mathbb{S}) is

Ext BP *(BP)(BP *,BP *), Ext_{BP_\ast(BP)}(BP_\ast, BP_\ast) \,,

where BPBP denotes the Brown-Peterson spectrum at prime pp.

recalled e.g. as Ravenel, theorem 1.4.2

References

The original articles are

  • John Adams, On the structure and applications of the Steenrod algebra, Comm. Math. Helv. 32 (1958), 180–214.

  • Sergei Novikov, The methods of algebraic topology from the viewpoint of cobordism theories, Izv. Akad. Nauk. SSSR. Ser. Mat. 31 (1967), 855–951 (Russian).

Reviews include for instance

Revised on November 17, 2013 06:54:34 by Urs Schreiber (89.204.130.133)