Eilenberg–Moore spectral sequence

Eilenberg–Moore spectral sequence The Eilenberg–Moore spectral sequence is a computational tool in Algebraic topology introduced by Samuel Eilenberg and John Coleman Moore in 1966 to calculate the homology or cohomology of pullbacks and fiber products. It refines the method of the Serre spectral sequence and connects to the Tor functor, the Bar construction, and derived functor techniques used in work by Jean-Pierre Serre, Henri Cartan, and Samuel Eilenberg's collaborators. The sequence plays a central role in calculations involving classifying spaces such as Eilenberg–MacLane spaces and spectra arising in the programs of Daniel Quillen and J. Peter May.

Introduction

The Eilenberg–Moore spectral sequence arises when studying a fibration f: X → B with fiber F and a map g: Y → B producing the pullback P = X ×_B Y; it relates the cohomology of X, Y, and B to that of P. Historically motivated by problems considered by Jean Leray and Jean-Pierre Serre, the construction leverages homological algebra developed by Samuel Eilenberg and Henri Cartan and builds on categorical perspectives later used by Saunders Mac Lane and Alexander Grothendieck. It is especially useful alongside the Serre spectral sequence, the Adams spectral sequence introduced by J. Frank Adams, and the homotopical tools developed by G. W. Whitehead and George W. Whitehead Jr..

Construction and Convergence

The algebraic input is the cobar or bar complex computing the derived tensor product (Tor) of differential graded algebras associated to the base and the maps from the total spaces; this uses constructions formalized by Daniel Quillen in his work on model categories and by Henri Cartan in earlier homological contexts. One forms a bicomplex from the normalized bar construction and filters it to obtain a first-quadrant spectral sequence whose E2-term is expressible in terms of Tor over the cohomology ring H*(B; k). Convergence results were established by Eilenberg and Moore under connectivity hypotheses analogous to those in the convergence criteria of Jean-Pierre Serre and later refined using model category arguments of Daniel Quillen and completion techniques explored by Ralph McKenzie. For rational coefficients the sequence often collapses, connecting to rational homotopy theory developed by Dennis Sullivan and calculations by Daniel Quillen.

Algebraic and Homotopical Framework

The Eilenberg–Moore spectral sequence is naturally formulated using differential graded algebra and model categories: one associates to maps into B differential graded modules over the differential graded algebra of cochains on B, as in frameworks used by J. Peter May and Mark Hovey. The E2-term is Tor_{H*(B)}(H*(X), H*(Y)), computed within the derived category techniques pioneered by Alexander Grothendieck and Jean-Louis Verdier. Homotopical refinements involve replacing spaces by simplicial sets, bringing in methods of Daniel Quillen and the simplicial machinery of André Joyal; this allows control of higher homotopies akin to those in the work of John Milnor and J. H. C. Whitehead. Connections with the Adams spectral sequence and the May spectral sequence appear when analyzing iterated fibrations or when computing the homotopy of function spaces studied by John F. Adams and J. P. May.

Computational Tools and Examples

Concrete computations employ the Eilenberg–Moore spectral sequence to compute H*(ΩX), the cohomology of loop spaces such as those studied by Raoul Bott and Bott periodicity, and to analyze fiber squares involving classifying spaces like BG for Lie groups Élie Cartan investigated in representation-theoretic contexts by Harish-Chandra. Classic examples include calculating the cohomology of the pullback of universal bundles over Grassmannians and computations in the study of configuration spaces addressed by Vladimir Arnold and Mikhail Gromov. Computational strategies often combine the spectral sequence with minimal model techniques of Dennis Sullivan and the algebraic machinery of Henri Cartan and Samuel Eilenberg, while modern implementations exploit computer algebra systems inspired by the work of John H. Conway and Donald Knuth in algorithm design.

Variants and Generalizations

Variants include versions for homology, for spectra in stable homotopy theory as developed by J. F. Adams and Haynes Miller, and equivariant forms applicable to actions of compact Lie groups studied by Atiyah–Bott and Michael Atiyah. Generalizations connect to the Rothenberg–Steenrod spectral sequence associated with classifying space fibrations considered by Marston Morse and to descent spectral sequences in the context of Alexander Grothendieck's descent theory. Extensions to structured ring spectra and higher category settings draw on work by Jacob Lurie and Paul Goerss and relate to derived algebraic geometry landscapes influenced by Maxim Kontsevich and Pierre Deligne.

Category:Algebraic topology