nLab
Serre 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

Contents

Idea

The Serre spectral sequence or Leray-Serre spectral sequence is a spectral sequence for computation of singular homology of topological spaces in a Serre-fiber sequence of topological spaces.

Given a homotopy fiber sequence

F E X \array{ F &\longrightarrow& E \\ && \downarrow \\ && X }

the the corresponding cohomology Serre spectral sequence looks like

E 2 p,q=H p(X,H q(F))H p+q(E). E_2^{p,q}= H^p(X, H^q(F)) \Rightarrow H^{p+q}(E) \,.

The generalization of this from ordinary cohomology to generalized (Eilenberg-Steenrod) cohomology is the Atiyah-Hirzebruch spectral sequence.

References

The original article is

  • Jean-Pierre Serre, Homologie singuliére des espaces fibrés Applications, Ann. of Math. 54 (1951),

A textbook account is for instance in

Lecture notes etc. includes

  • Greg Friedman, Some extremely brief notes on the Leray spectral sequence (pdf)

Discussion in homotopy type theory includes

Revised on May 7, 2015 13:45:08 by Urs Schreiber (195.113.30.252)