Lyndon spectral sequence

Lyndon spectral sequence
Name	Lyndon spectral sequence
Introduced	1950s
Author	Roger Lyndon
Field	Algebraic topology; Homological algebra; Group theory

Lyndon spectral sequence

The Lyndon spectral sequence is a computational tool in homological algebra and algebraic topology introduced in the study of group extensions and group cohomology. It relates filtered chain complexes arising from group presentations and exact couples to successive approximations of cohomology groups, enabling calculations across contexts linked to group extensions, modules, and crossed modules. It has influenced work connected to spectral sequences of Hochschild–Serre, Serre, Adams, and Bockstein and has connections with classical computations in homotopy theory.

Introduction

The Lyndon spectral sequence originates in work by Roger Lyndon and interacts with constructions considered by Jean-Pierre Serre, Claude Chevalley, Samuel Eilenberg, Saunders Mac Lane, and Kenneth Brown. It plays a role alongside the Hochschild–Serre spectral sequence, the Adams spectral sequence, the Bockstein spectral sequence, the Cartan–Eilenberg spectral sequence, and the Grothendieck spectral sequence in analyzing cohomological behavior of extensions and filtrations. Researchers such as John Milnor, Serge Lang, Hyman Bass, J. H. C. Whitehead, and Daniel Quillen have used related ideas in algebraic and homotopical settings. The construction is relevant to problems considered by Emmy Noether, David Hilbert, Alexander Grothendieck, Jean Leray, and René Thom in broader algebraic topology and homological algebra.

Construction

The foundational construction uses a filtered free resolution of a group or module akin to resolutions used by Emil Artin, Richard Brauer, Irving Kaplansky, and John Tate. Begin with a group presentation influenced by work of Otto Schreier and utilize a cellular chain complex approach in the tradition of H. S. M. Coxeter and Marston Morse. One forms an exact couple patterned after exact couples appearing in Jean Leray’s theory and applies methods reminiscent of Samuel Eilenberg and Saunders Mac Lane’s homological algebra. The filtration often arises from the lower central series studied by Wilhelm Magnus and Philip Hall or from augmentation ideals as in work by I. N. Herstein and Noam Elkies. The differentials and page structures echo constructions familiar from E. H. Brown and J. F. Adams style spectral sequences.

Convergence and E1/E2 Terms

Convergence results for the Lyndon spectral sequence parallel those proved by Jean Leray and Henri Cartan for other spectral sequences and connect with completeness phenomena explored by Alexander Grothendieck and Michael Artin. The E1 term typically encodes homology of subcomplexes constructed from graded pieces associated to a filtration, similar to the E1 pages in the Adams spectral sequence and in May spectral sequence contexts. The E2 term frequently recovers derived functor information analogous to Ext and Tor calculations championed by Emmy Noether and Samuel Eilenberg, and it can coincide with the familiar E2 term of the Hochschild–Serre spectral sequence under extension hypotheses studied by Claude Chevalley and Gerhard Hochschild. Convergence to associated graded pieces or to actual cohomology groups uses completeness and boundedness conditions familiar from work by Jean-Pierre Serre and Henri Cartan.

Applications in Group Cohomology

The Lyndon spectral sequence has been applied in computations of group cohomology of extensions considered by Claude Chevalley and Takashi Sugano and to analyze crossed modules introduced by J. H. C. Whitehead and Ronald Brown. It aids in determining low-dimensional cohomology classes and obstruction theories used by William Browder, Michael Atiyah, and Raoul Bott in obstruction and characteristic class contexts. In arithmetic settings linked to Emil Artin and Alexander Grothendieck it contributes to Galois cohomology computations touching work of Jean-Pierre Serre and Kurt Gödel’s contemporaries in logic-inspired algebra. It also appears in the study of cohomological dimension problems pursued by Serge Lang and Kenneth Brown and in analyses of group extensions treated by Otto Schreier and Philip Hall.

Examples and Computations

Concrete computations using the Lyndon spectral sequence include classical examples for free groups, surface groups, and one-relator groups examined in the tradition of Max Dehn and Jakob Nielsen. Calculations for nilpotent quotients reflect techniques of Wilhelm Magnus and Philip Hall, while computations for p-groups echo methods associated with John Milnor and Serre. Examples connecting to classifying spaces involve constructions by J. H. C. Whitehead and Daniel Quillen and relate to computations in algebraic K-theory championed by Quillen and Daniel Quillen’s collaborators. Case studies often reference computations by Hermann Weyl, Élie Cartan, and Norbert Wiener where filtered resolutions make explicit the E1 and E2 pages.

Variants and Generalizations

Several variants and generalizations parallel developments like the Hochschild–Serre spectral sequence and the Grothendieck spectral sequence and incorporate derived categorical perspectives due to Alexander Grothendieck and Henri Cartan. Extensions to filtered differential graded algebras invoke ideas from Jean-Michel Bismut and Daniel Quillen, while equivariant generalizations connect with work by Michael Atiyah and Friedrich Hirzebruch. Modern homotopical reforms employ model category techniques advanced by Daniel Quillen, Mark Hovey, and J. P. May, and derived functor reinterpretations draw on frameworks of Jacob Lurie and Bernhard Keller. Recent trends link to calculations in stable homotopy informed by Frank Adams, Douglas Ravenel, and J. H. Smith.

Category:Spectral sequences