# nLab Serre spectral sequence

### Context

#### Homological algebra

homological algebra

Introduction

diagram chasing

# Contents

## Idea

### In ordinary cohomology

The Serre spectral sequence or Leray-Serre spectral sequence is a spectral sequence for computation of ordinary cohomology (ordinary homology) of topological spaces in a Serre-fiber sequence of topological spaces.

Given a homotopy fiber sequence

$\array{ F &\longrightarrow& E \\ && \downarrow^{\mathrlap{p}} \\ && X }$

over a simply connected space $X$, then the corresponding cohomology Serre spectral sequence looks like

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

### In generalized cohomology

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

### In relative cohomology

There are two kinds of relative Serre spectral sequences.

For $F \to E \to X$ as above and $A \hookrightarrow X$ a subspace, the induced restriction of the fibration

$\array{ F & \simeq & F \\ \downarrow && \downarrow \\ p^{-1}(A) &\longrightarrow& E \\ \downarrow && \downarrow^{\mathrlap{p}} \\ A &\hookrightarrow& X }$

induces a spectral sequence in relative cohomology of the base space of the form

$E_2^{p,q} = H^p(X,A; H^q(F)) \;\Rightarrow\; H^\bullet(E, p^{-1}(A)) \,.$

(e.g. Davis 91, theorem 9.33)

Conversely, for

$\array{ F' & \hookrightarrow & F \\ \downarrow && \downarrow \\ E' &\hookrightarrow& E \\ \downarrow && \downarrow^{\mathrlap{p}} \\ X &\hookrightarrow& X }$

a sub-fibration over the same base, then this induces a spectral sequence for relative cohomology of the the total space in terms of ordinary cohomology with coefficients in the relative cohomology of the fibers:

$E^{p,q}_2 = H^p(X; H^q(F,F')) \;\Rightarrow\; H^\bullet(E,E') \,.$

### In equivariant cohomology

There is also a generalization to equivariant cohomology: for cohomology with coefficients in a Mackey functor withRO(G)-grading for representation spheres $S^V$, then for $E \to X$ an $F$-fibration of topological G-spaces and for $A$ any $G$-Mackey functor, the equivariant Serre spectral sequence looks like (Kronholm 10, theorem 3.1):

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

where on the left in the $E_2$-page we have ordinary cohomology with coefficients in the genuine equivariant cohomology groups of the fiber.

## Details

For details on the plain Serre spectral sequence see at Atiyah-Hirzebruch spectral sequence and take $E = H R$ to be ordinary cohomology.

## References

### General

The original article is

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

Textbook accounts include

Lecture notes etc. includes

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

Discussion in homotopy type theory includes

### In equivariant cohomology

• Ieke Moerdijk, J.-A. Svensson, The Equivariant Serre Spectral Sequence, Proceedings of the American Mathematical Society Vol. 118, No. 1 (May, 1993), pp. 263-278 (JSTOR)

and for genuine equivariant cohomology, i.e. for RO(G)-graded cohomology with coefficients in a Mackey functor:

• William Kronholm, The $RO(G)$-graded Serre spectral sequence, Homology Homotopy Appl. Volume 12, Number 1 (2010), 75-92. (pdf, Euclid)