Contents

# 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.

###### Proposition

(ordinary cohomology Serre spectral sequence)

Given a homotopy fiber sequence

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

over a connected topological space $X$, such that the canonical group action of the fundamental group $\pi_1(X)$ on the ordinary cohomology of the fiber $F$ is trivial (for instance if $X$ is a simply connected topological space), then there exists a cohomology Serre spectral sequence of the form:

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

(e.g. Hatcher, Thm. 1.14)

Hence for $n \in \mathbb{N}$ we have a filtration of abelian groups

(2)$0 \xhookrightarrow{ \;\; E^{n,0}_\infty \;\; } F^n_n \xhookrightarrow{ E^{n-1,1}_\infty } F^n_{n-1} \xhookrightarrow{\;\;\;\;\;} \cdots \xhookrightarrow{\;\;\;\;\;} F^n_{1} \xhookrightarrow{ \;\; E^{0,n}_\infty \;\; } F^n_0 \;=\; H^n(E) \,,$

where

$F^n_{p+1} \xhookrightarrow{ E^{p,n-p}_\infty } F^n_{p} {\phantom{AAAA}} \text{means that} {\phantom{AAAA}} 0 \to F^n_{p+1} \xhookrightarrow{\;} F^n_{p} \twoheadrightarrow E^{p,n-p}_\infty \to 0 \,,$

hence that – iteratively as $p$ decreases$F^n_p$ is an extension of $E^{p,n-p}_\infty$ by $F^n_{p+1}$.

### 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.

## Examples

###### Example

Let

Then the integral cohomology in degree 4 of the homotopy quotient is the direct sum

(3)$H^4 \big( S^{\mathbb{H}} \!\sslash\! G, \, \mathbb{Z} \big) \;\; \simeq \;\; \mathbb{Z} \oplus (\mathbb{Z}/\left\vert G \right\vert)$

of the integers with the cyclic group of order that of $G$.

###### Proof

By the Borel construction we have a homotopy fiber sequence of the form

(4)$\array{ S^4 &\longrightarrow& S^{\mathbb{H}} \!\sslash\! G \\ && \big\downarrow \\ && B G }$

over the classifying space of $G$.

Here the integral cohomology of the 4-sphere fiber is (e.g. by the nature of the Eilenberg-MacLane space $K(\mathbb{Z},4)$)

(5)$H^n \big( S^4, \, \mathbb{Z} \big) \;\simeq\; \left\{ \array{ \mathbb{Z} & for & n \in \{0,4\} \\ 0 & \text{otherwise} \,. } \right.$

We claim that the group action of $\pi_1(B G) \simeq G$ (by this Prop.) on the integral cohomology of the fiber is trivial. This follows by observing that:

1. we have an isomorphism of topological G-spaces between the representation sphere of $\mathbb{H}$ and the unit sphere in $\mathbb{R} \oplus \mathbb{H}$ (by this Prop.):

$S^{\mathbb{H}} \;\simeq_{G}\; S(\mathbb{R} \oplus \mathbb{H}) \,;$
2. the group action of Sp(1) on $\mathbb{H} \simeq_{\mathbb{R}} \mathbb{R}^4$ is through the defining action of SO(4), hence the action on $\mathbb{R} \oplus \mathbb{H}$ is through SO(5),

because quaternions are a normed division algebra, so that left-multiplication by unit-norm quaternions $q \in$ Sp(1) $= S(\mathbb{H})$ preserves the norm (e.g HSS 18, Rem. A.8);

3. the generator of $H^4(S^4,\mathbb{Z})$ may be identified with the volume form (under the Hopf degree theorem and the de Rham theorem) which is manifestly preserved by the action of the special orthogonal group $SO(5)$.

Therefore, the integral-cohomological Serre spectral sequence (Prop. ) applies to the Borel fiber sequence (4).

Now, noticing that the integral cohomology of a classifying space of a discrete group is its group cohomology

$H^\bullet(B G, \mathbb{Z}) \;\simeq\; H^\bullet_{grp}(G, \mathbb{Z})$

we have for the given finite subgroup of SU(2) (by this Prop) that:

(6)$H^n(B G, \mathbb{Z}) \;\simeq\; \left\{ \array{ \mathbb{Z} &\vert& n = 0 \\ G^{ab} &\vert& n = 2 \, mod \, 4 \\ \mathbb{Z}/{\vert G \vert} &\vert& n \, \text{positive multiple of} \, 4 \\ 0 &\vert& \text{otherwise} \,, } \right.$

where $G^{ab} \coloneqq G / [G,G]$ denotes the abelianization of $G$.

Using the cohomology groups (5) and (6) in the fomula (1) for the second page $E_2^{\bullet, \bullet}$ of the cohomology Serre spectral sequence (Prop. ) shows that this is of the following form:

Since the codomains of the differentials on all the following pages are translated diagonally (downwards and rightwards, by the general formula) from the codomains seen above, one sees that for every differential on every page, the domain or the codomain is the zero group.

This means that all differentials are the zero morphism, hence that the spectral sequence collapses already on this second page:

$E^{\bullet, \bullet}_\infty \;\; \simeq \;\; E^{\bullet, \bullet}_2 \,.$

Therefore the convergence statement (2) says that the degree-4 cohomology group in question is a group extension of

$E^{0,4}_\infty \;\simeq\; H^0\big(B G, \, H^4(S^4; \mathbb{Z}) \big) \;\simeq\; H^0\big( B G, \, \mathbb{Z}\big) \;\simeq\; \mathbb{Z}$

by

$E^{4,0}_\infty \;\simeq\; H^4\big(B G, \, H^0(S^4; \mathbb{Z}) \big) \;\simeq\; H^4\big( B G, \, \mathbb{Z}\big) \;\simeq\; \mathbb{Z}/\left\vert G \right\vert$

in that we have a short exact sequence of the form

$0 \to \mathbb{Z}/\left\vert G\right\vert \hookrightarrow H^4\big( S^4 \!\sslash\! G;\, \mathbb{Z} \big) \twoheadrightarrow \mathbb{Z} \to 0 \,.$

But since the Ext-group of the integers is trivial (this Expl.) this extension must be the direct sum

$H^4\big( S^4 \!\sslash\! G;\, \mathbb{Z} \big) \;\simeq\; \mathbb{Z}_{\left\vert G\right\vert} \oplus \mathbb{Z} \,.$

This is the claim (3) to be proven.

## References

### General

The original article is

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

Textbook accounts:

Lecture notes etc. includes

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

Discussion in homotopy type theory includes

and implementation in Lean is in

### 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)