# nLab Serre spectral sequence

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# 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

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

the the corresponding cohomology Serre spectral sequence looks like

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

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)