# Serre spectral sequence

The Serre spectral sequence (also called the Leray–Serre spectral sequence) is an algebraic tool of topology that computes the homology or cohomology of the total space of a fibration \( F \to E \to B \) from the homology of the base \( B \) and the fiber \( F \), together with information about how the base twists the fiber.

| Key fact | Detail |
|---|---|
| Input | A Serre fibration \( F \to E \to B \); the sequence relates \( H_*(F) \), \( H_*(B) \), and \( H_*(E) \)<sup>[2](https://www.sciencedirect.com/topics/mathematics/spectral-sequence)</sup> |
| \( E^2 \) term (homology) | \( E^2_{p,q} \cong H_p(B;\,{}^{t}H_q(F)) \), where \( {}^{t}H_*(F) \) is the local coefficient system encoding the action of \( \pi_1(B) \) on the fiber<sup>[2](https://www.sciencedirect.com/topics/mathematics/spectral-sequence)</sup> |
| Simple case | If \( B \) is simply connected and \( \mathrm{Tor}(H_*(B), H_*(F)) = 0 \), then \( E^2_{p,q} \cong H_p(B) \otimes H_q(F) \)<sup>[2](https://www.sciencedirect.com/topics/mathematics/spectral-sequence)</sup> |
| Convergence | It is a first quadrant spectral sequence converging to \( H_*(E) \), so convergence is not an issue<sup>[2](https://www.sciencedirect.com/topics/mathematics/spectral-sequence)</sup> |
| Cohomology version | \( E_2^{p,q} \cong H^p(B;\,{}^{t}H^q(F)) \Rightarrow H^{p+q}(E) \); it is a spectral sequence of algebras, with differentials satisfying the Leibniz rule for the cup product<sup>[2](https://www.sciencedirect.com/topics/mathematics/spectral-sequence)</sup> |
| Serre exact sequence | If \( B \) is simply connected with \( H_i(B) = 0 \) for \( i \le n \) and \( H_j(F) = 0 \) for \( j \le m \), there is a long exact sequence \( H_k(F) \to H_k(E) \to H_k(B) \to H_{k-1}(F) \to \cdots \) valid for \( k < m+n \)<sup>[3](https://mathweb.ucsd.edu/~ebelmont/231br-notes.pdf)</sup> |
| Generalization | The construction works for any homology theory when the base is a finite-dimensional CW complex; the identity fibration \( X \to X \) yields the Atiyah–Hirzebruch spectral sequence<sup>[4](https://pi.math.cornell.edu/~hatcher/SSAT/SSch1.pdf)</sup> |

## What a spectral sequence does

A spectral sequence is a sequence of pages \( E_r \) of abelian groups arranged on a grid with coordinates \( (p, q) \), where \( r \) is the page number. Each page carries differentials \( d_r : E^r_{p,q} \to E^r_{p-r,\,q+r-1} \), and the next page is obtained from the previous one as kernel modulo image<sup>[3](https://mathweb.ucsd.edu/~ebelmont/231br-notes.pdf)</sup>. One way to obtain such an object is from a chain of inclusions of chain complexes \( E^0_* \subset E^1_* \subset \cdots \subset E^N_* \): the snake lemma gives a long exact sequence for a single inclusion, and iterating the construction for a whole filtration leads to a spectral sequence<sup>[3](https://mathweb.ucsd.edu/~ebelmont/231br-notes.pdf)</sup>. Equivalently, every exact couple yields a spectral sequence, and every filtered complex yields an exact couple<sup>[5](https://www.mathi.uni-heidelberg.de/~mfuetterer/texts/specseq.pdf)</sup>.

For the Serre spectral sequence specifically, a CW structure on the base gives a geometric filtration of the total space, and the exact couple built from the homology of the filtration stages \( D^1_{p,q} = H_{p+q}(F_p X) \) and \( E^1_{p,q} = H_{p+q}(F_p X, F_{p-1} X) \) produces the sequence. Theorems on fibrations guarantee that this filtration is cocomplete and that \( E^1_{p,q} = 0 \) if either \( p < 0 \) or \( q < 0 \), so the Serre spectral sequence is always a first quadrant spectral sequence converging to \( H_*(E) \)<sup>[2](https://www.sciencedirect.com/topics/mathematics/spectral-sequence)</sup>.

## The E2 term and local coefficients

The identification of the \( E^2 \) page is Serre's theorem: for the fibration \( F \to E \to B \), there is an isomorphism

\[ E^2_{p,q} \cong H_p(B;\,{}^{t}H_q(F)), \]

where \( {}^{t}H_*(F) \) denotes a twisted, or local, coefficient system in which the differential is modified to account for the action, coming from the fibration, of the fundamental groupoid of \( B \) on the fiber \( F \)<sup>[2](https://www.sciencedirect.com/topics/mathematics/spectral-sequence)</sup>. Concretely, each element of \( \pi_1 B \) gives a homotopy class of maps \( F \to F \), and hence a map \( H^*(F; R) \to H^*(F; R) \); this construction defines the action of \( \pi_1 B \) on \( H^* F \)<sup>[6](https://strickland1.org/courses/bestiary/ss.pdf)</sup>.

The local coefficients disappear under familiar hypotheses. If \( B \) is simply connected and \( \mathrm{Tor}(H_*(B), H_*(F)) = 0 \), the universal coefficient theorem reduces the \( E^2 \) term to \( E^2_{p,q} \cong H_p(B) \otimes H_q(F) \)<sup>[2](https://www.sciencedirect.com/topics/mathematics/spectral-sequence)</sup>. Over a field \( K \), if \( \pi_1(B) = 0 \) or acts trivially on the fiber cohomology and \( B \) and \( F \) are of finite type, then \( E_2^{p,q} = H^p(B; K) \otimes_K H^q(F; K) \)<sup>[7](https://people.math.wisc.edu/~lmaxim/754notes.pdf)</sup>. More generally, if \( \pi_1 B \) acts trivially on \( H^*(F; R) \), the \( E^2 \) terms are just \( H^s(B; H^t(F; R)) \)<sup>[6](https://strickland1.org/courses/bestiary/ss.pdf)</sup>.

## The cohomological version and multiplicative structure

The cohomological Serre spectral sequence has the form

\[ E_2^{p,q} \cong H^p(B;\,{}^{t}H^q(F)) \Rightarrow H^{p+q}(E). \]

Its advantage is that it is a spectral sequence of algebras. Each page carries a product induced from the cup product on \( H^*(B) \) and \( H^*(F) \), and the differentials \( d_r \) satisfy the graded Leibniz rule

\[ d_r(x \cdot y) = d_r(x) \cdot y + (-1)^{\deg x}\, x \cdot d_r(y), \]

with \( \deg(x) = p + q \)<sup>[2](https://www.sciencedirect.com/topics/mathematics/spectral-sequence)</sup><sup> • </sup><sup>[7](https://people.math.wisc.edu/~lmaxim/754notes.pdf)</sup>. This restriction greatly simplifies the computation of differentials and also allows computation of the cup product on \( H^*(E) \)<sup>[2](https://www.sciencedirect.com/topics/mathematics/spectral-sequence)</sup>.

The edge maps are also determined by the maps of the fibration. When \( \pi_1(B) = 0 \) or acts trivially on the fiber cohomology, the compositions \( H^q(B) = E_2^{q,0} \twoheadrightarrow \cdots \twoheadrightarrow E_\infty^{q,0} \hookrightarrow H^q(E) \) and \( H^q(E) \twoheadrightarrow E_\infty^{0,q} \hookrightarrow \cdots \hookrightarrow E_2^{0,q} = H^q(F) \) are precisely the homomorphisms \( \pi^* : H^q(B) \to H^q(E) \) and \( i^* : H^q(E) \to H^q(F) \) induced by the projection and the fiber inclusion<sup>[7](https://people.math.wisc.edu/~lmaxim/754notes.pdf)</sup>.

## The Serre exact sequence

A common use of the sequence produces a long exact sequence in homology. Suppose \( B \) is simply connected with \( H_i(B) = 0 \) for \( i \le n \) and \( H_j(F) = 0 \) for \( j \le m \). Then the \( E^2 \) page is empty on the rectangle \( [0,n] \times [0,m] \) except at \( (0,0) \), and the same holds for \( E^\infty \). In that range one obtains a long exact sequence

\[ \cdots \to H_k(F) \to H_k(E) \to H_k(B) \to H_{k-1}(F) \to \cdots \]

valid for \( k < m + n \)<sup>[3](https://mathweb.ucsd.edu/~ebelmont/231br-notes.pdf)</sup>. This is called the Serre exact sequence. It says that, in a large range of dimensions, \( B \) looks like the mapping cone of \( F \to E \); in this range cofibrations and fibrations behave alike<sup>[3](https://mathweb.ucsd.edu/~ebelmont/231br-notes.pdf)</sup>.

## Leray–Hirsch and splitting

The spectral sequence also explains when the cohomology of the total space splits as a tensor product. The Leray–Hirsch theorem states: if \( F \to E \to B \) is a fibration with \( B \) and \( F \) of finite type, \( \pi_1(B) = 0 \), and the fiber inclusion induces a surjection \( i^* : H^*(E; K) \to H^*(F; K) \) over a field \( K \), then \( H^*(E; K) \cong H^*(B; K) \otimes_K H^*(F; K) \)<sup>[7](https://people.math.wisc.edu/~lmaxim/754notes.pdf)</sup>. The proof runs through the spectral sequence: surjectivity of \( i^* \) forces all differentials restricted to the \( q \)-axis to vanish, the Leibniz rule then kills every differential, and over a field all extension problems from \( E_\infty \) to \( H^*(E; K) \) are trivial<sup>[7](https://people.math.wisc.edu/~lmaxim/754notes.pdf)</sup>.

## A worked example

The fibration \( S^1 \to S^{2n+1} \to \mathbb{CP}^n \) illustrates the machinery. With

\[ H^*S^1 = \mathbb{Z}[u]/u^2, \quad |u| = 1; \qquad H^*\mathbb{CP}^n = \mathbb{Z}[x]/x^{n+1}, \quad |x| = 2, \]

the \( E_2 \) page is \( \mathbb{Z}[u,x]/(u^2, x^{n+1}) \). The differential satisfies \( d_2(u) = x \), and by the Leibniz rule \( d_2(x^k u) = x^{k+1} \) while \( d_2(x^k) = 0 \). It follows that \( E_3 = \mathbb{Z}\{1, ux^n\} \), there are no further differentials, and \( E_3 = E_\infty \), which matches \( H^*S^{2n+1} = \mathbb{Z}[v]/v^2 \) with \( |v| = 2n+1 \)<sup>[6](https://strickland1.org/courses/bestiary/ss.pdf)</sup>. The same fibration shifted to \( S^{2n+1} \to \mathbb{CP}^n \to \mathbb{CP}^\infty \) gives a spectral sequence \( \mathbb{Z}[x] \otimes E[u] \Rightarrow H^*\mathbb{CP}^\infty \)<sup>[6](https://strickland1.org/courses/bestiary/ss.pdf)</sup>.

## Directions of use and generalizations

In applications one frequently works backwards. Instead of computing \( H_*(E) \) from known \( H_*(F) \) and \( H_*(B) \), it is often \( H_*(E) \) and one of the other two groups that are known, and the spectral sequence is run in reverse to find the homology of the third space<sup>[2](https://www.sciencedirect.com/topics/mathematics/spectral-sequence)</sup>.

The construction extends well beyond ordinary homology. For a general homology theory \( h_* \), the Serre spectral sequence works provided the base \( B \) is a finite-dimensional [CW complex](https://www.edgepedia.org/cw-complex). Taking the fibration to be the identity map \( X \to X \) yields a spectral sequence converging to \( h_*(X) \) with \( E^2_{p,q} = H_p(X; h_q(\mathrm{point})) \), the Atiyah–Hirzebruch spectral sequence, together with its cohomology analog<sup>[4](https://pi.math.cornell.edu/~hatcher/SSAT/SSch1.pdf)</sup>. There is also a relative version \( H^s(B, B'; H^t(F; R)) \Rightarrow H^{s+t}(E, E'; R) \) for a subspace \( B' \subset B \) with \( E' = q^{-1}B' \), which carries a module structure over the absolute sequence<sup>[6](https://strickland1.org/courses/bestiary/ss.pdf)</sup>. The Leray–Serre and Eilenberg–Moore spectral sequences, both fundamental for computing the cohomology of a space, admit a joint tri-graded refinement relating the two when they share the same abutment<sup>[8](https://ems.press/journals/dm/articles/14298378)</sup>.

## References

1. [Serre, Jean-Pierre. "Homologie singulière des espaces fibrés." Annals of Mathematics 54 (1951), 425–505; MIT OpenCourseWare 18.906 references](https://ocw.mit.edu/courses/18-906-algebraic-topology-ii-spring-2020/pages/references/)
2. [Spectral Sequence – an Overview, ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/spectral-sequence)
3. [Notes on algebraic topology, Lecture 7 (E. Belmont, UCSD)](https://mathweb.ucsd.edu/~ebelmont/231br-notes.pdf)
4. [Hatcher, Spectral Sequences in Algebraic Topology, Chapter 1: The Serre Spectral Sequence](https://pi.math.cornell.edu/~hatcher/SSAT/SSch1.pdf)
5. [Fütterer, Spectral Sequences, University of Heidelberg](https://www.mathi.uni-heidelberg.de/~mfuetterer/texts/specseq.pdf)
6. [Strickland, Spectral Sequences (course notes)](https://strickland1.org/courses/bestiary/ss.pdf)
7. [Maxim, Lecture notes on homotopy theory and applications, University of Wisconsin](https://people.math.wisc.edu/~lmaxim/754notes.pdf)
8. [Neumann, Frank, and Szymik, Markus. "Preludes to the Eilenberg–Moore and the Leray–Serre spectral sequences." Doc. Math. 29 (2024), 1319–1339](https://ems.press/journals/dm/articles/14298378)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Homological algebra and K-theory › Spectral sequences and homological techniques*

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