Adams–Novikov spectral sequence

Adams–Novikov spectral sequence
Name	Adams–Novikov spectral sequence
Field	Algebraic topology
Introduced by	J. F. Adams, S. P. Novikov
Year	1960s
Related	Adams spectral sequence, Brown–Peterson cohomology, Morava K-theory, Chromatic homotopy theory

Adams–Novikov spectral sequence is a spectral sequence used in stable homotopy theory to compute stable homotopy groups of spheres and related spectra by means of complex-oriented cohomology theories and formal group laws. It refines the classical Adams spectral sequence by replacing ordinary cohomology with Brown–Peterson cohomology, complex cobordism, or other extraordinary cohomology theories, providing access to deeper periodic phenomena discovered by researchers such as Haynes Miller, Douglas Ravenel, and J. F. Adams. The sequence connects algebraic invariants like Ext groups over the Hopf algebroid of cooperations to topological computations central to chromatic homotopy theory, Morava stabilizer group, and the work of Sergei Novikov and collaborators.

Introduction

The Adams–Novikov spectral sequence arises from the interplay between complex cobordism MU, the Lazard ring, and the algebra of cooperations MU_*MU studied by Milnor, Quillen, and Novikov. Its E_2-term is expressed as Ext over a Hopf algebroid (MU_*, MU_*MU), linking to structural results by Daniel Quillen, John Milnor, and J. F. Adams. The tool was developed in the milieu of breakthroughs by Sergei Novikov on the Novikov conjecture and the work of Milnor on cobordism theory, and later integrated into the program of Ravenel and Miller on periodicity and localizations formulated by Bousfield and Ravenel.

Construction and algebraic setup

Construction uses a complex-oriented spectrum such as MU or BP and its cooperations MU_*MU or BP_*BP, treated as a Hopf algebroid following ideas of Ravenel, Landweber, and Johnson–Wilson. One forms a cobar complex whose homology yields Ext groups Ext_{(MU_*,MU_*MU)}^{s,t}(MU_*, MU_*X) for a spectrum X, invoking homological algebra techniques from Cartan–Eilenberg, Künneth theorem contexts, and computational frameworks established by Adams, Novikov, and Quillen. The algebraic input frequently uses the Lazard ring and formal group law classification by Quillen, while change-of-rings theorems of Landweber and the Ravenel conjectures facilitate passage between MU and BP, invoking modules over Steenrod algebra analogues and the action of the Morava stabilizer group.

Computation and convergence

Convergence issues involve conditional convergence and strong convergence results proved using Boardman, Bousfield, and J. F. Adams techniques, with counterexamples and subtleties addressed by Margolis and Miller. Computational strategies exploit vanishing lines, periodicity operators from Morava K-theory, and localizations at primes p following Ravenel and Bousfield. Spectral sequence differentials are analyzed using families discovered by Toda, Mahowald, and Miller–Wilson methods, while hidden extensions are resolved by appealing to exotic elements described by Novikov, Johnson–Wilson, and calculations in the Adams–Novikov E_2-term by Ravenel. The role of profinite groups such as the Morava stabilizer group and cohomology of Lubin–Tate spaces figures in convergence proofs and the computation of continuous cohomology.

Relationship to other spectral sequences

The Adams–Novikov spectral sequence generalizes and refines the Adams spectral sequence and connects to the Bockstein spectral sequence, Atiyah–Hirzebruch spectral sequence, and chromatic spectral sequences appearing in the work of Ravenel, Hopkins, Smith, and Sadofsky. Change-of-rings comparisons relate it to Ext computations over the Steenrod algebra studied by Adams and Milnor, and to the May spectral sequence and the bar spectral sequence in homological algebra developed by May and Eilenberg–Mac Lane. The interplay with Morava K-theory and Lubin–Tate theory situates it within the machinery employed by Hopkins, Miller, Devinatz, and Goerss.

Applications in stable homotopy theory

Applications include computation of stable homotopy groups of spheres and complex-oriented spectra, detection of periodic families such as vn-periodicity studied by Ravenel and Smith, and structural results in chromatic homotopy theory by Hopkins, Ravenel, and Lurie. It underpins work on the Nilpotence theorem and the Thick subcategory theorem by Devinatz–Hopkins–Smith, informs calculations in topological modular forms (tmf) studied by Hopkins and Mahowald, and plays a role in analyses of the Adams–Novikov differentials relevant to the solution of the Kervaire invariant problem pursued by Hill–Hopkins–Ravenel.

Examples and notable computations

Notable computations include classical work on the stable homotopy groups of spheres at chromatic heights by Novikov, Adams, Ravenel, and Toda, BP-based charts computed by Boardman and Ravenel–Wilson–Wilson, and modern computations using Morava E-theory by Hopkins–Miller and Goerss–Hopkins–Miller. Explicit charts and vanishing lines at primes such as 2, 3, and odd primes appear in papers by Mahowald, Adams–Priddy, and Bruner, while recent advances connect to computations in topological Hochschild homology and algebraic K-theory by Hesselholt and Madsen.

Category:Algebraic topology