---
tag: AHW4B5480K
kind: statement
role: theorem
name: Serre spectral sequence
msc: [55T10, 55R20]
uses: [PB9JXQ2KC3, M58WRZ7PFM, 6N7BC70CX0, S1XK3HXHFX, D3BN7MBKTR]
about: [PB9JXQ2KC3]
sources:
  - url: https://en.wikipedia.org/wiki/Serre_spectral_sequence
same_as:
  wikidata: Q7455410
  wikipedia: Serre spectral sequence
---

## Statement

Let $\pi \colon E \to B$ be a [[Serre fibration|PB9JXQ2KC3]] with $B$ path-connected, let $b_0 \in B$, and let $F = \pi^{-1}(b_0)$ be the fibre.
Let $R$ be a commutative ring.
Then there is a first-quadrant cohomological spectral sequence $(E_r^{p,q}, d_r)_{r \ge 2}$, with $d_r \colon E_r^{p,q} \to E_r^{p+r, q-r+1}$, such that:

1. $E_2^{p,q} = H^p(B; \mathcal{H}^q(F; R))$, the [[singular cohomology|6N7BC70CX0]] of $B$ with [[local coefficients|M58WRZ7PFM]] in the system $b \mapsto H^q(\pi^{-1}(b); R)$;
2. the spectral sequence converges to $H^{p+q}(E; R)$: there is a filtration of $H^n(E; R)$ whose successive quotients are the groups $E_\infty^{p, n-p}$;
3. it is multiplicative: each $E_r$ is a bigraded ring and $d_r$ is a derivation.
On $E_2^{p,q} \otimes E_2^{s,t}$ the product is the cup product times $(-1)^{qs}$.

If the [[fundamental group|S1XK3HXHFX]] $\pi_1(B, b_0)$ acts as the identity on $H^*(F; R)$, for instance if $B$ is simply connected, then $E_2^{p,q} = H^p(B; H^q(F; R))$ with constant coefficients.

## Remarks

One proof filters the singular cochains of $E$ by the preimages of the skeleta of a CW approximation of $B$.
Another proof filters the singular simplices of $E$ by how they project to $B$.
There is a homological version with $E^2_{p,q} = H_p(B; \mathcal{H}_q(F; R))$ converging to $H_{p+q}(E; R)$.

Example: for $n \ge 2$, the [[path space fibration|D3BN7MBKTR]] $\Omega S^n \to P S^n \to S^n$ has contractible total space.
So all classes except $E_2^{0,0}$ must die, which forces $d_n \colon E_n^{0, k(n-1)} \to E_n^{n, (k-1)(n-1)}$ to be isomorphisms.
It follows that $H^i(\Omega S^n; \Z) \cong \Z$ when $n - 1$ divides $i$ and $0$ otherwise.

## Explanations

### L1 human

A fibration is a space $E$ that looks locally like a product of a base $B$ and a fibre $F$.
The Serre spectral sequence starts from the cohomology of $B$ with coefficients in the cohomology of $F$, as if $E$ were a product.
Then it corrects this guess step by step, and the end result is the cohomology of $E$.

### L2 human

The Serre spectral sequence is the main tool for computing cohomology of fibre bundles, loop spaces, homogeneous spaces and Eilenberg–MacLane spaces.
It gives the Gysin and Wang sequences as special cases when the fibre or the base is a sphere.
Serre used it to prove that the homotopy groups of spheres are finitely generated, and finite except in a few degrees.