Serre spectral sequence

Serre spectral sequence
Name	Serre spectral sequence
Field	Algebraic topology
Introduced	1951
Introduced by	Jean-Pierre Serre

Serre spectral sequence

The Serre spectral sequence is an algebraic tool in Algebraic topology introduced by Jean-Pierre Serre to compute homology and cohomology of fiber bundles such as fibrations and fiber bundles. It connects the homotopy and homology of the base, total space, and fiber, and plays a central role in calculations related to Hurewicz, Eilenberg–MacLane spaces, Postnikov towers, and classifications in obstruction theory.

Introduction

The sequence arises from a filtration of the singular chain complex of a total space in a fibration whose base is a CW complex or a space satisfying conditions similar to those in Hurewicz and Whitehead contexts; it was developed by Jean-Pierre Serre during investigations linked to Eilenberg–MacLane spaces, Allen Hatcher-style expositions, and later expository treatments by authors associated with Princeton University, Harvard University, and Institute for Advanced Study. Influences include methods from Henri Cartan, Samuel Eilenberg, Norman Steenrod, and ideas seen in the work of Élie Cartan and Maurice Fréchet.

Construction and statement

Given a Serre fibration F → E → B with base B a CW complex or a space meeting Serre class conditions, one constructs a filtration of the Singular chain complex by subcomplexes indexed by the skeleta of B, a method related to constructions by Leray, Grothendieck in spectral contexts, and classical filtrations used in Hochschild–Serre analogues. The associated graded object yields a first quadrant spectral sequence whose E2-page is expressed as E2^{p,q} ≅ H^p(B; H^q(F; R)) for cohomology with coefficients in a ring R, paralleling patterns in Lyndon and Grothendieck frameworks, and converges to H^{p+q}(E; R) under conditions like simple connectivity or nilpotence appearing in works by Serre, Dennis Sullivan, and Daniel Quillen.

Examples and computations

Classical computations include the cohomology of loop spaces ΩX, where the spectral sequence applied to the path-loop fibration relates to computations in Eilenberg–MacLane spaces and calculations by Serre on homotopy groups of spheres invoked in the proof of results related to the Bott periodicity theorem and Adams spectral sequence comparisons. Other examples: the fibration S^1 → S^{2n+1} → CP^n gives computations tied to complex projective space cohomology classes and Chern classes studied by Shiing-Shen Chern; the Hopf fibration S^1 → S^3 → S^2 relates to calculations important in Hopf and analyses by Hurewicz and Freudenthal. Computations for principal G-bundles over manifolds draw from structure theory of Lie groups like SU(n), SO(n), and Sp(n), connecting to characteristic classes investigated by Chern, Lev Pontryagin and Eduard Stiefel collaborators.

Properties and convergence

The spectral sequence is a first quadrant spectral sequence with differentials d_r : E_r^{p,q} → E_r^{p+r,q-r+1} reminiscent of patterns in the Adams and Atiyah–Hirzebruch machines studied by Michael Atiyah and Friedrich Hirzebruch, and its convergence properties depend on connectivity hypotheses familiar from Serre and nilpotent space conditions invoked in work by Quillen and Sullivan. Under simple connectivity of B and finite type conditions, the spectral sequence strongly converges to the graded pieces of H^*(E; R), and edge homomorphisms recover maps induced by inclusion and projection that are core to analyses in Hurewicz and Whitehead contexts.

Applications in topology

Applications include computations of homology and cohomology of classifying spaces such as BG for a Lie group G, interactions with characteristic classes in bundle classification problems central to Thom and Pontryagin analyses, and use in modern approaches to rational homotopy theory developed by Sullivan and Quillen. The tool is deployed in studies of manifold invariants considered by John Milnor, Sergei Novikov, and in computations relevant to K-theory as in work by Atiyah, Bott, and Graeme Segal.

Variants and generalizations

Variants include the homology version with E2_{p,q} ≅ H_p(B; H_q(F; R)), nonabelian generalizations appearing in obstruction-theoretic contexts linked to Postnikov towers and Eilenberg–Moore, and extensions to equivariant settings associated with Borel and equivariant cohomology frameworks used in studies by Atiyah, Segal, and Armand Borel. Further generalizations connect to spectral sequences in homological algebra such as the Grothendieck and Cartan–Eilenberg methods employed by Eilenberg and Cartan.

Category:Algebraic topology