Algebraic topology Spectral sequences Theorem

Serre spectral sequence

Builds on Serre fibration Local coefficient system Singular cohomology Fundamental group Path space fibration

Statement

Let $\pi \colon E \to B$ be a Serre fibration 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:

$E_2^{p,q} = H^p(B; \mathcal{H}^q(F; R))$ , the singular cohomology of $B$ with local coefficients in the system $b \mapsto H^q(\pi^{-1}(b); R)$ ;
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 $E_\infty^{p, n-p}$ ;
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 $\pi_1(B, b_0)$ 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 $\Omega S^n \to P S^n \to S^n$ 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.

Going deeper

For a student

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.

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Wang sequence

Tag	`AHW4B5480K`
URI	`m2://AHW4B5480K`
Kind	statement, theorem
Writing	W0 stub
MSC	55T10, 55R20
Status	This copy of the site has no build data. Run `m2 status AHW4B5480K`.
About	Serre fibration
Uses	Serre fibration, Local coefficient system, Singular cohomology, Fundamental group, Path space fibration
Same as	Wikidata Q7455410 Wikipedia Serre spectral sequence
Changed	2 November 2026

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Wang sequence