03.13.02 · modern-geometry / spectral-sequences

Leray-Serre spectral sequence and the Gysin sequence

shipped3 tiersLean: partialpending prereqs

Anchor (Master): Serre 1951 PhD thesis; Bott-Tu §14–§17; McCleary §5–§6; Borel *Topology of Lie groups*

Intuition [Beginner]

When a space is built as a fibration — a base space $B$ with a copy of a fibre $F$ glued over each point — its cohomology is approximately the cohomology of $B$ multiplied by the cohomology of $F$ . The Leray-Serre spectral sequence is the bookkeeping device that turns this approximation into an exact computation, page by page.

The starting page $E_{2}$ records the naive product: $E_{2}^{p, q} = H^{p} (B; H^{q} (F))$ . If the fibration is a product, this is the answer, by the Künneth formula. If the fibration is twisted, the higher pages of the spectral sequence introduce corrections that account for the twist. The cohomology of the total space $E$ is whatever survives after all corrections.

The simplest twisted example is the Hopf fibration $S^{1} \to S^{3} \to S^{2}$ . The naive product $H^{*} (S^{2}) \cdot H^{*} (S^{1})$ has cohomology in degrees $0, 1, 2, 3$ . But $S^{3}$ has cohomology only in degrees $0$ and $3$ . The Leray-Serre spectral sequence corrects the over-count: a single non-zero differential on the $E_{2}$ page pairs the class in degree $1$ with the class in degree $2$ and erases both, leaving exactly the cohomology of $S^{3}$ .

Visual [Beginner]

Picture a grid for the Hopf fibration. The horizontal axis is $p$ (degree on the base $S^{2}$ ), the vertical axis is $q$ (degree on the fibre $S^{1}$ ). The grid has non-zero entries only at $(0, 0), (0, 1), (2, 0), (2, 1)$ — the four corners of a $2 \times 1$ rectangle.

The differential $d_{2}$ on the $E_{2}$ page goes from $(0, 1)$ to $(2, 0)$ — diagonally down-and-right by $(2, - 1)$ . It maps the fibre class to the Euler class of the bundle, which on the Hopf fibration is the generator of $H^{2} (S^{2})$ . After $d_{2}$ cancels the source and target, the surviving entries on $E_{3} = E_{\infty}$ are $(0, 0)$ and $(2, 1)$ , with total degrees $0$ and $3$ — matching the cohomology of $S^{3}$ .

Worked example [Beginner]

The Hopf fibration computation. Start with $E_{2}^{p, q} = H^{p} (S^{2}; H^{q} (S^{1}))$ . Both $H^{p} (S^{2})$ and $H^{q} (S^{1})$ are one-dimensional in degrees $0$ and $2$ (resp. $0$ and $1$ ), so the $E_{2}$ grid is:

$E_{2}^{0, 0} = Z$
$E_{2}^{0, 1} = Z$
$E_{2}^{2, 0} = Z$
$E_{2}^{2, 1} = Z$
All other entries are zero.

The differential $d_{2} : E_{2}^{0, 1} \to E_{2}^{2, 0}$ is multiplication by the Euler class of the Hopf bundle, which equals $1$ in $H^{2} (S^{2}) = Z$ . So $d_{2}$ is the identity, and the cohomology of $E_{2}$ on the second page kills both source and target.

After this differential, $E_{3} = E_{\infty}$ has only $E_{\infty}^{0, 0} = Z$ and $E_{\infty}^{2, 1} = Z$ . The total degrees are $0$ and $3$ . The cohomology of $S^{3}$ is $Z$ in degrees $0$ and $3$ and zero elsewhere — match.

This computation is impossible without the spectral sequence: there is no Mayer-Vietoris cover of $S^{3}$ that exhibits its cohomology directly through the Hopf structure. The spectral sequence is the only tool that uses the fibre-bundle data.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $π : E \to B$ be a fibration of CW complexes with fibre $F = π^{- 1} (b_{0})$ over a basepoint $b_{0} \in B$ , with $B$ simply connected (the general case requires a local-system formulation).

Theorem (Serre 1951). There is a first-quadrant cohomological spectral sequence with $$ E_2^{p, q} = H^p(B; H^q(F)) $$ converging to $H^{p + q} (E)$ as a graded ring (multiplicatively, when $E_{r}$ inherits a multiplicative structure).

Construction. The skeletal filtration $F^{p} E := π^{- 1} (B^{(p)})$ , where $B^{(p)}$ is the $p$ -skeleton of $B$ , is a decreasing filtration of $E$ by closed subspaces. Applying singular cohomology produces a filtration on the singular cochain complex $C^{*} (E)$ by setting $F^{p} C^{*} (E) =$ kernel of restriction $C^{*} (E) \to C^{*} (F^{p - 1} E)$ . This is a bounded filtration, and the filtered-complex spectral sequence of 03.13.01 applies. The identification of the $E_{2}$ page as $H^{p} (B; H^{q} (F))$ uses the Serre fibration property — that the associated graded $F^{p} E / F^{p - 1} E$ is a wedge of $(F, b_{0})$ -pairs indexed by the $p$ -cells of $B$ .

Differential bidegree. The page- $r$ differential is $d_{r} : E_{r}^{p, q} \to E_{r}^{p + r, q - r + 1}$ (notation decision #15). The first non-vanishing differential is $d_{2} : E_{2}^{0, 1} \to E_{2}^{2, 0}$ , which on the Hopf fibration corresponds to multiplication by the Euler class.

Multiplicative structure. The cup product on singular cochains is compatible with the skeletal filtration ( $F^{p} ⌣ F^{p^{'}} \subset F^{p + p^{'}}$ ), so each page $E_{r}$ is a bigraded ring and $d_{r}$ is a derivation. On $E_{2}$ , the product is the cup product $H^{p} (B; H^{q} (F)) \otimes H^{p^{'}} (B; H^{q^{'}} (F)) \to H^{p + p^{'}} (B; H^{q + q^{'}} (F))$ induced from the cup product on $H^{*} (F)$ . The associated graded of $H^{*} (E)$ as a ring is $E_{\infty}$ ^{[Bott-Tu §14, multiplicative structure]}.

Edge homomorphisms. The bottom edge $E_{\infty}^{p, 0} \subset E_{2}^{p, 0} = H^{p} (B)$ supplies the fibre-restriction part of the bundle map; explicitly, the composition $$ H^p(B) = E_2^{p, 0} \twoheadrightarrow E_\infty^{p, 0} \hookrightarrow H^p(E) $$ is the bundle pullback $π^{*} : H^{p} (B) \to H^{p} (E)$ . The left edge supplies the fibre-restriction: $$ H^q(E) \twoheadrightarrow E_\infty^{0, q} \hookrightarrow E_2^{0, q} = H^q(F) $$ is the restriction to the fibre.

Transgression. A class $x \in E_{2}^{0, q}$ that survives to $E_{q}$ — i.e., is killed by no $d_{r}$ for $r \leq q$ — has a (possibly non-zero) differential $d_{q + 1} (x) \in E_{q + 1}^{q + 1, 0} \subset H^{q + 1} (B)$ , called the transgression of $x$ . Equivalently, the transgression is the composition $H^{q} (F) ⇢ H^{q + 1} (B)$ defined where it makes sense, satisfying the cocycle identity in the Leray construction.

Gysin sequence. When $F = S^{r - 1}$ is a sphere, the cohomology of the fibre is concentrated in degrees $0$ and $r - 1$ : $$ E_2^{p, 0} = H^p(B), \qquad E_2^{p, r-1} = H^p(B; H^{r-1}(S^{r-1})) = H^p(B), $$ with all other entries zero (assuming the bundle is orientable so the local system on $H^{r - 1} (F)$ is constant). The only non-vanishing differential is $d_{r} : E_{r}^{p, r - 1} \to E_{r}^{p + r, 0}$ , which by transgression is multiplication by the Euler class $e \in H^{r} (B)$ . The resulting long exact sequence is $$ \cdots \to H^{p-r}(B) \xrightarrow{\smile e} H^p(B) \xrightarrow{\pi^} H^p(SE) \xrightarrow{\pi_} H^{p-r+1}(B) \xrightarrow{\smile e} H^{p+1}(B) \to \cdots $$ where $π_{*} : H^{*} (S E) \to H^{*- r + 1} (B)$ is integration along the fibre. This is the Gysin sequence ^{[Gysin 1942]}.

Key theorem with proof [Intermediate+]

Theorem (Hopf fibration via Leray-Serre). The Leray-Serre spectral sequence of the Hopf fibration $S^{1} ι S^{3} π S^{2}$ has $E_{2} = E_{3} = \dots = E_{\infty}$ except for a single non-zero differential $d_{2} : E_{2}^{0, 1} \to E_{2}^{2, 0}$ , which is multiplication by $\pm 1$ . The resulting $E_{\infty}$ matches $H^(S^3)$.*

Proof. With $Z$ coefficients, $$ E_2^{p, q} = H^p(S^2; H^q(S^1)) = \begin{cases} \mathbb{Z} & (p, q) \in {(0, 0), (0, 1), (2, 0), (2, 1)}, \ 0 & \text{else}. \end{cases} $$ The differential $d_{2} : E_{2}^{p, q} \to E_{2}^{p + 2, q - 1}$ has bidegree $(2, - 1)$ and is non-zero only on the source $E_{2}^{0, 1} = Z$ , which it maps to $E_{2}^{2, 0} = Z$ . This $d_{2}$ is the transgression of the fibre fundamental class $[S^{1}] \in H^{1} (S^{1})$ , which by the Serre construction equals the Euler class $e (η) \in H^{2} (S^{2})$ of the Hopf bundle.

The Hopf bundle has Euler class equal to the generator of $H^{2} (S^{2}) = Z$ (the bundle is the universal $U (1)$ -bundle over $C P^{1} = S^{2}$ , with first Chern class $1$ ). Hence $d_{2} : Z \to Z$ is multiplication by $\pm 1$ , an isomorphism. Both source and target are killed.

After $d_{2}$ , the surviving entries on $E_{3} = E_{\infty}$ are:

$E_{\infty}^{0, 0} = Z$
$E_{\infty}^{2, 1} = Z$
All others zero.

Total degrees: $0$ and $2 + 1 = 3$ . Matching $H^{*} (S^{3}) = Z$ in degrees $0$ and $3$ . The associated graded of $H^{*} (S^{3})$ is exactly $E_{\infty}$ . $□$

The same computation generalises to the Hopf bundles $S^{1} \to S^{2 n + 1} \to C P^{n}$ . The $E_{2}$ page is $H^{*} (C P^{n}) \otimes H^{*} (S^{1})$ as a bigraded module, and the only non-zero differential $d_{2}$ is multiplication by the Euler class $c_{1}$ of the tautological line bundle. The cohomology after $d_{2}$ is concentrated in total degrees $0$ and $2 n + 1$ , recovering $H^{*} (S^{2 n + 1})$ .

Theorem (Gysin sequence from Leray-Serre). For an oriented sphere bundle $S^{r - 1} \to S E π B$ of total space $S E$ , there is a long exact sequence $$ \cdots \to H^{p-r}(B) \xrightarrow{\smile e(E)} H^p(B) \xrightarrow{\pi^} H^p(SE) \xrightarrow{\pi_} H^{p-r+1}(B) \to \cdots $$ where $e (E) \in H^{r} (B)$ is the Euler class.

Proof sketch. Run the Leray-Serre spectral sequence. The fibre cohomology is $H^{*} (S^{r - 1}) = Z$ in degrees $0$ and $r - 1$ , zero elsewhere. The $E_{2}$ page has only two non-zero rows: $q = 0$ and $q = r - 1$ . The differential of bidegree $(r, 1 - r)$ from $q = r - 1$ row to $q = 0$ row is the only possible non-zero differential (other differentials land outside the support). This differential is multiplication by the Euler class $e (E)$ , the transgression of the fibre fundamental class. Splicing the kernel-cokernel pieces of this single differential into a long exact sequence gives the Gysin sequence. The maps $π^{*}$ and $π_{*}$ identify with the edge homomorphisms.

Synthesis. The Leray-Serre spectral sequence is exactly the filtered-complex spectral sequence of a fibration. This is precisely Serre's 1951 specialisation of the abstract Leray apparatus. The Gysin sequence is a special case of Leray-Serre on an oriented sphere bundle.

Bridge. The construction here builds toward 03.13.03 (leray-hirsch theorem and the splitting principle for vector bundles), where the same data is developed in the next layer of the strand. The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Exercise 3 (medium, symbolic). Gysin on $C P^{k}$ .

Apply the Gysin sequence to the unit-circle bundle $S^{1} \to S^{2 k + 1} \to C P^{k}$ (the unit-sphere bundle of $O (1)$ ). Use the resulting sequence to derive $H^{*} (C P^{k}) = Z [x] / (x^{k + 1})$ , $de g x = 2$ .

Hint

The Euler class of $O (1)$ over $C P^{k}$ is the generator $x$ of $H^{2} (C P^{k})$ .

Answer

The Gysin sequence with $r = 2$ , $e = x$ (Euler class of $O (1)$ ): $$ \cdots \to H^{p-2}(\mathbb{C}P^k) \xrightarrow{\smile x} H^p(\mathbb{C}P^k) \to H^p(S^{2k+1}) \to H^{p-1}(\mathbb{C}P^k) \xrightarrow{\smile x} H^{p+1}(\mathbb{C}P^k) \to \cdots $$ We know $H^{*} (S^{2 k + 1}) = Z$ in degrees $0$ and $2 k + 1$ . Inductively: $H^{0} (C P^{k}) = Z$ from the bottom of the sequence; $H^{1} = 0$ (since $H^{1} (S^{2 k + 1}) = 0$ forces $H^{1} (C P^{k}) \to H^{0} (C P^{k}) ⌣ x H^{2} (C P^{k})$ injective in the relevant slot); proceeding up by two, multiplication by $x$ is an isomorphism in non-vanishing degrees up to $2 k$ , and zero past $2 k$ . Thus $H^{2 j} (C P^{k}) = Z$ generated by $x^{j}$ for $0 \leq j \leq k$ , and $H^{2 j - 1} = 0$ . As a ring, $Z [x] / (x^{k + 1})$ — the relation $x^{k + 1} = 0$ comes from the Gysin sequence in degree $2 k + 1$ , where multiplication by $x$ kills the top class.

Exercise 4 (medium, proof). Section-induced collapse.

Let $F \to E \to B$ be a fibration that admits a section $s : B \to E$ with $π s = id_{B}$ , and assume $B$ is simply connected. Show that the Leray-Serre spectral sequence collapses at $E_{2}$ .

Hint

The section gives a splitting of the bundle pullback $π^{*} : H^{*} (B) \to H^{*} (E)$ , hence injectivity of $π^{*}$ on cohomology.

Answer

The bundle pullback $π^{*} : H^{*} (B) \to H^{*} (E)$ is the bottom edge homomorphism of the spectral sequence; it factors as $H^{*} (B) = E_{2}^{*, 0} ↠ E_{\infty}^{*, 0} ↪ H^{*} (E)$ . The composition $s^{*} π^{*} = id^{*} = id$ on $H^{*} (B)$ shows that $π^{*}$ is split injective.

This means the surjection $E_{2}^{*, 0} ↠ E_{\infty}^{*, 0}$ is an isomorphism, i.e., no class in the bottom row $q = 0$ is killed by any $d_{r}$ . Hence no differential of bidegree $(r, 1 - r)$ originating in row $q \geq 1$ can land in row $q = 0$ (the target is non-zero, and there is nothing to map onto, but classes in row $q = 0$ are not in the image of any non-zero $d_{r}$ either).

But we need also that no class in row $q = 0$ is the source of a non-zero $d_{r}$ , which is automatic: $d_{r}$ from row $q = 0$ goes to $q = - r + 1 < 0$ , outside the first quadrant. Combined with multiplicativity (each $d_{r}$ is a derivation, so a $d_{r}$ -cycle on the bottom row generates $d_{r}$ -cycles by multiplication), the entire spectral sequence collapses at $E_{2}$ . $□$

Exercise 5 (medium, symbolic). Borel computation of $H^(BU(1))$.*

Use the path-loop fibration $Z \to R \to S^{1}$ — which deloops to $K (Z, 1) \to P K (Z, 2) \to K (Z, 2)$ with $K (Z, 1) = S^{1}$ and $P K (Z, 2)$ contractible — to compute $H^{*} (K (Z, 2)) = H^{*} (B U (1))$ .

Hint

The path space is contractible. The Serre spectral sequence of the path-loop fibration $Ω X \to P X \to X$ converges to $H^{*} (pt)$ , so all classes except in $E_{\infty}^{0, 0}$ must die.

Answer

For $X = K (Z, 2) = C P^{\infty}$ , the Serre SS of $S^{1} \to P K (Z, 2) \to K (Z, 2)$ has $E_{2} = H^{*} (K (Z, 2)) \otimes H^{*} (S^{1})$ . Convergence to $H^{*} (pt) = Z$ in degree $0$ forces all higher classes to be killed.

The only non-vanishing differential is $d_{2} : E_{2}^{0, 1} \to E_{2}^{2, 0}$ . Let $σ \in H^{1} (S^{1})$ be the generator and $x = d_{2} (σ) \in H^{2} (K (Z, 2))$ . Then $x$ generates $H^{2}$ . By the multiplicative structure, $d_{2} (σ \cdot x^{k}) = x^{k + 1}$ , so $x^{k + 1}$ is in the image of $d_{2}$ and survives only if $x^{k + 1} \neq = 0$ but is hit, which means $x^{k + 1}$ generates $H^{2 k + 2}$ . Inductively, $H^{2 k} (K (Z, 2)) = Z \cdot x^{k}$ for all $k \geq 0$ , and odd degrees vanish.

Thus $H^{*} (K (Z, 2)) = Z [x]$ with $de g x = 2$ , recovering $H^{*} (C P^{\infty}) = H^{*} (B U (1))$ .

Exercise 6 (medium, proof). Wang sequence.

For a fibration $F \to E \to S^{n}$ with $n \geq 2$ and $F$ connected, derive the Wang exact sequence $$ \cdots \to H^p(E) \to H^p(F) \xrightarrow{\theta} H^{p-n+1}(F) \to H^{p+1}(E) \to \cdots $$ where $θ$ is the "twisting" derivation.

Hint

The base $S^{n}$ has cohomology only in degrees $0$ and $n$ . The $E_{2}$ page has only two non-zero columns.

Answer

$E_{2}^{p, q} = H^{p} (S^{n}; H^{q} (F))$ . Since $H^{p} (S^{n}) = 0$ except for $p = 0$ and $p = n$ , only columns $p = 0$ and $p = n$ are non-zero. The only possible differential is $d_{n} : E_{n}^{0, q} \to E_{n}^{n, q - n + 1}$ .

The cohomology of $E$ is built from kernel of $d_{n}$ in column $0$ plus cokernel of $d_{n}$ landing in column $n$ . Splicing together, the long exact sequence $$ 0 \to E_\infty^{n, q - n} \to H^q(E) \to E_\infty^{0, q} \to 0 $$ combined with $E_{\infty}^{0, q} = ker d_{n} \subset H^{q} (F)$ and $E_{\infty}^{n, q - n} = H^{q - n} (F) / im (d_{n})$ gives the Wang sequence. The map $θ = d_{n} : H^{q} (F) \to H^{q - n + 1} (F)$ is the connecting morphism. $□$

Exercise 7 (hard, symbolic). Cohomology of $Ω S^{n}$ .

Use the Serre SS of the path-loop fibration $Ω S^{n} \to P S^{n} \to S^{n}$ (with $P S^{n}$ contractible) to compute $H^{*} (Ω S^{n}; Q)$ for $n \geq 2$ .

Hint

Reverse engineer: $H^{*} (P S^{n}) = Q$ in degree $0$ only, so all $E_{\infty}$ entries except $(0, 0)$ must die. Use the multiplicative structure to determine $H^{*} (Ω S^{n})$ inductively.

Answer

For $n$ even, $E_{2}^{p, q} = H^{p} (S^{n}) \otimes H^{q} (Ω S^{n}; Q)$ . Non-zero only in columns $p = 0$ and $p = n$ . The differential $d_{n} : E_{n}^{0, q} \to E_{n}^{n, q - n + 1}$ must kill all classes outside $E_{\infty}^{0, 0}$ . Let $σ \in H^{n - 1} (Ω S^{n})$ be the transgression source ( $d_{n} (σ) =$ fundamental class of $S^{n}$ ). By multiplicativity, $d_{n} (σ^{k}) = k σ^{k - 1} \cdot d_{n} (σ)$ , and rationally this is non-zero for all $k \geq 1$ . Hence $H^{*} (Ω S^{n}; Q) = Q [σ] \otimes Λ [τ]$ with $de g σ = 2 (n - 1)$ , $de g τ = n - 1$ , polynomial in even-degree generators and exterior in odd-degree. Specifically $H^{*} (Ω S^{2 m}; Q) = Λ [τ_{2 m - 1}] \otimes Q [σ_{4 m - 2}]$ for $n = 2 m$ even, and $H^{*} (Ω S^{2 m + 1}; Q) = Q [τ_{2 m}]$ for $n = 2 m + 1$ odd. (The odd case is purely polynomial because $τ \cdot τ \neq = 0$ rationally on $Ω S^{odd}$ .)

Exercise 8 (hard, proof). Multiplicative collapse.

Suppose $F \to E \to B$ is a fibration with $B$ simply connected, $H^{*} (F; k)$ free over a field $k$ , and there exist classes $e_{1}, \dots, e_{n} \in H^{*} (E; k)$ whose restrictions $ι^{*} (e_{i}) \in H^{*} (F; k)$ form a $k$ -vector-space basis. Prove the Leray-Hirsch theorem: $H^{*} (E; k)$ is a free $H^{*} (B; k)$ -module with basis ${e_{i}}$ .

Hint

The classes $e_{i}$ supply lifts of the $E_{2}$ -row $E_{2}^{0, *}$ to $H^{*} (E)$ . Show that the spectral sequence collapses at $E_{2}$ by induction on filtration degree.

Answer

Define a homomorphism $Φ : H^{*} (B; k) \otimes H^{*} (F; k) \to H^{*} (E; k)$ by $Φ (β \otimes ι^{*} e_{i}) = π^{*} (β) \cdot e_{i}$ . We show $Φ$ is an isomorphism, equivalently that the spectral sequence collapses at $E_{2}$ .

Filter both sides by $π^{*}$ -degree: on the left, by powers of the augmentation ideal of $H^{*} (B; k)$ ; on the right, by skeletal filtration. $Φ$ respects filtration. The induced map on associated gradeds is the identification $E_{2}^{p, q} = H^{p} (B; H^{q} (F)) \to gr^{p} H^{p + q} (E)$ .

If $Φ$ is surjective on associated gradeds, the multiplicative structure forces all $d_{r}$ to vanish: any $d_{r} e_{i} \in E_{r}^{r, q - r + 1}$ would be in the image of $Φ$ , hence would already be a cycle, contradiction.

Surjectivity on associated gradeds: for any $α \in E_{2}^{p, q}$ , write $α = \sum β_{j} \cdot ι^{*} e_{i_{j}}$ in $H^{p} (B; H^{q} (F)) = H^{p} (B; k) \otimes H^{q} (F; k)$ using the basis ${ι^{*} e_{i}}$ . The lift $\sum π^{*} (β_{j}) \cdot e_{i_{j}}$ in $H^{p + q} (E; k)$ has image $α$ in $gr^{p} H^{p + q}$ . Hence $Φ$ is a graded isomorphism, the spectral sequence collapses at $E_{2}$ , and $H^{*} (E; k) ≅ H^{*} (B; k) \otimes H^{*} (F; k)$ as a graded $H^{*} (B; k)$ -module. $□$

Lean formalization [Intermediate+]

[object Promise]

The Leray-Serre spectral sequence is one of the largest gaps in Mathlib's algebraic-topology infrastructure. The path forward requires Serre fibrations (partially present), bigraded cohomology (partial), the skeletal filtration (absent), and the spectral-sequence convergence theorem applied to this filtration (absent).

Advanced results [Master]

Cohomology of the loop space. The Serre spectral sequence of the path-loop fibration $Ω X \to P X \to X$ , with $P X$ contractible, computes $H^{*} (Ω X)$ from $H^{*} (X)$ . The reverse engineering — knowing the abutment is concentrated in degree zero forces all classes on $E_{\infty}$ except $E_{\infty}^{0, 0}$ to vanish — recovers $H^{*} (Ω S^{n})$ rationally and partially $H^{*} (Ω K (A, n))$ . This is the principal computational route to cohomology of loop spaces and to Cartan's tables of cohomology of Eilenberg-MacLane spaces.

Borel's theorem. For a compact Lie group $G$ with maximal torus $T$ and Weyl group $W$ , the Serre spectral sequence of the fibration $G / T \to B T \to B G$ degenerates and identifies $H^{*} (B G; Q)$ with the Weyl-group invariants of $H^{*} (B T; Q)$ : $$ H^(BG; \mathbb{Q}) \cong H^(BT; \mathbb{Q})^W. $$ For $G = U (n)$ , $T = (S^{1})^{n}$ , $W = S_{n}$ : $H^{*} (B T) = Q [x_{1}, \dots, x_{n}]$ , so $H^{*} (B U (n); Q) = Q [x_{1}, \dots, x_{n}]^{S_{n}}$ , the symmetric polynomials in the Chern roots — recovering $H^{*} (B U (n)) = Q [c_{1}, \dots, c_{n}]$ . This is the foundation of the modern theory of characteristic classes ^{[Borel 1955]}.

Serre's finiteness. The Serre SS of the Whitehead tower $K (π_{n} (X), n) \to X ⟨ n ⟩ \to X ⟨ n - 1 ⟩$ controls how homotopy groups assemble into cohomology. Serre's 1953 Groupes d'homotopie et classes de groupes abéliens used this machinery to prove that $π_{k} (S^{n})$ is finite for $k > n$ (except for $π_{n} (S^{n}) = Z$ and $π_{4 n - 1} (S^{2 n})$ which has $Z$ summand from the Hopf invariant). The proof: rational cohomology of $S^{n}$ is concentrated in degrees $0$ and $n$ ; rationally, the Whitehead-tower spectral sequence kills all higher homotopy modulo torsion. The integer statement requires p-local refinement.

Gysin and Euler. The Euler class $e (E) \in H^{r} (B; Z)$ of a rank- $r$ oriented vector bundle, originally defined as the Poincaré dual of the zero section of a generic section (when one exists), is identified by the Leray-Serre spectral sequence with the transgression of the fibre fundamental class in the Serre SS of the unit-sphere bundle $S^{r - 1} \to S E \to B$ . This is the only spectral-sequence-derivation route to the Euler class that works for non-spin bundles and arbitrary base; the Gysin sequence is its computational shadow.

Eilenberg-Moore spectral sequence. Dual to Leray-Serre, the Eilenberg-Moore spectral sequence computes $H^{*} (F)$ for the homotopy fibre $F$ of a map $f : X \to Y$ , with $$ E_2^{p, q} = \operatorname{Tor}{H^(Y)}^{p, q}(H^(X), \mathbb{Z}) ;\Rightarrow; H^{p+q}(F). $$ The convergence is more delicate (requires simply-connectedness of $Y$ and finitely-generated homology in each degree) but the computation is sometimes more tractable than the Serre SS; e.g., it recovers $H^{*} (Ω S^{n})$ via $E_2 = \operatorname{Tor}{H^*(S^n)}(\mathbb{Z}, \mathbb{Z})$ in a single page.

Multiplicative structure and Steenrod operations. The Serre SS is compatible with Steenrod operations $Sq^{i}$ on $F_{2}$ -cohomology: each page is a module over $A_{2}$ , and the differentials commute with $Sq^{i}$ . This compatibility, due to Kudo and refined by Adams, makes the Serre SS the workhorse for computing $Sq^{i}$ -action on classifying-space cohomology.

Full proof set [Master]

Construction of the Serre spectral sequence — full account. Let $π : E \to B$ be a Serre fibration with $B$ a CW complex with skeleta $B^{(p)}$ . Define $$ F^p E := \pi^{-1}(B^{(p)}), \qquad F^p E \subset F^{p+1} E \subset \cdots \subset E. $$ This is an increasing filtration of $E$ by closed subspaces. The corresponding decreasing filtration on the singular cochain complex $C^{*} (E; G)$ for a coefficient group $G$ is $$ F^p C^(E; G) := \ker!\bigl(C^(E; G) \to C^(F^{p-1} E; G)\bigr). $$ Boundedness ($F^0 = C^(E; G) $an d$ F^{n+1} = 0 $in co h o m o l o g i c a l d e g r ee$ n $) f o l l o w s f r o m t h ece l l u l a r s t r u c t u r eo f$ B$.

The associated graded is identified by the Serre fibration property: the relative cohomology $H^{*} (F^{p} E, F^{p - 1} E; G)$ is the cohomology of a wedge over $p$ -cells of $B$ of $(F, b_{0})$ -pairs, and by the Eilenberg-Steenrod axioms equals $$ H^(F^p E, F^{p-1} E; G) \cong C^p(B; H^(F; G)), $$ the cellular cochains of $B$ with coefficients in the cohomology of the fibre.

The exact couple from this filtration has $E_{1}^{p, q} = H^{p + q} (F^{p} E, F^{p - 1} E; G) = C^{p} (B; H^{q} (F; G))$ and $d_{1} = δ$ the cellular coboundary. Hence $$ E_2^{p, q} = H^p(B; H^q(F; G)). $$ The spectral sequence converges to $H^{*} (E; G)$ by 03.13.01.

Multiplicativity from cup products. The cup product on $C^{*} (E; R)$ (for a ring $R$ ) restricts to $F^{p} ⌣ F^{p^{'}} \subset F^{p + p^{'}}$ because $π^{- 1} (B^{(p)} \cap B^{(p^{'})}) \subset π^{- 1} (B^{(p + p^{'})})$ . The resulting bigraded multiplication on each page makes $d_{r}$ a derivation by the universal property of derived couples on a multiplicative filtered DGA. The $E_{2}$ product is the cup product on $H^{*} (B; H^{*} (F; R))$ induced by the cup product $H^{*} (F; R) \otimes H^{*} (F; R) \to H^{*} (F; R)$ .

Edge homomorphism = bundle pullback. The bottom-row inclusion $E_{\infty}^{p, 0} ↪ E_{2}^{p, 0} = H^{p} (B)$ identifies with $π^{*} : H^{p} (B) \to H^{p} (E)$ as follows. The bottom row $E_{\infty}^{p, 0}$ is the image in $H^{p} (E)$ of classes pulled back from $B$ ; the surjection $H^{p} (B) = E_{2}^{p, 0} ↠ E_{\infty}^{p, 0}$ is induced by the differentials $d_{r}$ which kill classes in $H^{p} (B)$ that are obstructed (i.e., not in the image of $π^{*}$ ). Combining the surjection and inclusion gives $π^{*}$ .

Edge homomorphism = fibre restriction. The left-column inclusion $E_{\infty}^{0, q} ↪ E_{2}^{0, q} = H^{q} (F)$ identifies with the fibre restriction $ι^{*} : H^{q} (E) \to H^{q} (F)$ , dualizing the bottom-row case.

Transgression. The transgression $τ : H^{q} (F) ⇢ H^{q + 1} (B)$ is the partial map defined on classes that survive to $E_{q}$ in column $0$ and lands in column $q + 1$ via $d_{q + 1}$ . In the Hopf fibration, the fibre fundamental class $[S^{1}] \in H^{1} (S^{1})$ transgresses to the Euler class of the bundle in $H^{2} (S^{2})$ . The Gysin sequence is the long exact sequence of transgression in a sphere bundle.

Gysin sequence — explicit derivation. Take $F = S^{r - 1}$ . The cohomology $H^{q} (S^{r - 1}) = Z$ for $q = 0$ and $q = r - 1$ , zero elsewhere. Hence $E_{2}^{p, q}$ is non-zero only for $q = 0$ and $q = r - 1$ . The only possibly non-zero differential is $d_{r} : E_{r}^{p, r - 1} \to E_{r}^{p + r, 0}$ .

By the universal Euler-class computation, $d_{r}$ on the column $p = 0$ row $q = r - 1$ is the transgression $$ d_r([S^{r-1}]) = e(E) \in H^r(B), $$ and by multiplicativity $d_{r} (x \cdot α) = x \cdot e (E)$ for $α \in H^{p} (B) = E_{2}^{p, 0}$ and $x \in E_{2}^{0, r - 1}$ , i.e., $d_{r}$ on $E_{2}^{p, r - 1}$ is multiplication by $e (E) : H^{p} (B) \to H^{p + r} (B)$ .

The page $E_{r + 1}$ has $$ E_{r+1}^{p, 0} = H^p(B) / e(E) \cdot H^{p-r}(B), \qquad E_{r+1}^{p, r-1} = \ker(e(E) \smile : H^p(B) \to H^{p+r}(B)). $$ After this, no further differentials can act (they would land outside the support), so $E_{\infty} = E_{r + 1}$ . The two-row filtration on $H^{n} (S E)$ then splices into the long exact sequence $$ \cdots \to E_\infty^{n, 0} \to H^n(SE) \to E_\infty^{n - r + 1, r - 1} \to 0 $$ combined with $E_{\infty}^{n - r + 1, r - 1} ↪ H^{n - r + 1} (B) e (E) ⌣ H^{n + 1} (B)$ , giving the Gysin sequence.

Connections [Master]

Spectral sequence (general) 03.13.01 — Leray-Serre is the geometric specialisation. By conn:441.serre-finiteness, Serre spectral sequence is the filtered-complex SS of a fibration's singular cochain filtration (specialisation). The exact-couple machinery of the upstream unit is what powers this construction.
Leray-Hirsch theorem 03.13.03 — the collapse-at- $E_{2}$ specialisation of Leray-Serre. When fibre cohomology classes extend to total-space classes, the spectral sequence collapses on page two and the abutment is the Künneth product. The downstream unit develops this and the splitting principle.
Pontryagin and Chern classes 03.06.04 — Borel's identification $H^{*} (B G; Q) = H^{*} (B T; Q)^{W}$ via the Serre SS of $G / T \to B T \to B G$ is the structural foundation of the modern theory of characteristic classes. By conn:442.serre-gysin-euler, Gysin sequence + Euler class derived from Serre spectral sequence of an oriented sphere bundle (foundation-of). For an oriented $S^{r - 1}$ -bundle the Serre SS collapses at $E_{2}$ except for one differential of bidegree $(r, 1 - r)$ ; that differential is multiplication by the Euler class, recovering the Gysin LES.
Whitehead tower and homotopy of spheres 03.12.07 — by conn:443.serre-loop-space, Serre SS of path-loop fibration computes loop-space cohomology and π_n(S^k) (foundation-of). Serre 1951's path-loop fibration $Ω X \to P X \to X$ with $P X$ contractible gives an SS converging to a point, allowing $Ω X$ cohomology to be read off; $π_{4} (S^{3}) = Z /2$ is computed via this device. The same fibration recurs in the next bullet for $K (A, n)$ . By conn:450.serre-finiteness-pi-spheres, Finiteness of π_k(S^n) for k > n built on Whitehead tower and Serre SS (foundation-of); this is the Serre 1953 theorem proved by rational Hurewicz on the Whitehead tower.
Eilenberg-MacLane spaces 03.12.05 — the Serre SS of the path-loop fibration $Ω K (A, n) \to * \to K (A, n)$ supplies the inductive computation $H^{*} (K (A, n))$ from $H^{*} (K (A, n - 1))$ . This is the gateway to cohomology operations and the Steenrod algebra.
Atiyah-Singer index theorem 03.09.10 — Borel-Hirzebruch's spectral-sequence computation of $H^{*} (B U (n))$ supplies the Chern-character target for the topological side of Atiyah-Singer.
K-theory 03.08.01 — the Atiyah-Hirzebruch spectral sequence $H^{p} (X; K^{q} (pt)) \Rightarrow K^{p + q} (X)$ is the K-theoretic shadow of Leray-Serre, with the same skeletal-filtration construction applied to a generalised cohomology theory.
Singular cohomology and de Rham 03.04.13 — Leray-Serre is stated for singular cohomology; in the smooth fibre-bundle case, the de Rham version (Bott-Tu §14) gives the same answer with the differential-form complex of $E$ filtered by skeleta of $B$ .

The Hopf fibration calculation is the cleanest worked example. The Borel computation is the structural payload — it routes characteristic-class theory through the spectral sequence. The Gysin sequence is the unique route to the Euler class for non-spin bundles in low rank.

Throughlines and forward promises. Leray-Serre is the foundational tool for fibration cohomology. We will see Borel's identification $H^{*} (B G; Q) = H^{*} (B T; Q)^{W}$ run through this machine in 03.08.05; we will see $π_{4} (S^{3}) = Z /2$ computed via the Whitehead-tower fibration in 03.12.07; we will later see the family-index theorem of 03.09.21 use the equivariant Serre SS. This pattern recurs throughout characteristic-class theory and homotopy theory. The foundational reason the spectral sequence converges to total cohomology is exactly Serre's filtration of singular cochains by skeleta of the base. Putting these together: Leray-Serre is a specialisation of the abstract filtered-complex SS, an instance of the broader local-to-global computational paradigm, and the bridge between fibre/base data and total-space cohomology. The Gysin sequence is precisely the Serre SS of an oriented sphere bundle collapsed to its single non-zero differential — this is exactly the Euler-class transgression, and this pattern recurs whenever an oriented bundle appears.

Historical & philosophical context [Master]

Jean Leray introduced the spectral sequence of a continuous map in his 1946 announcement L'anneau d'homologie d'une représentation (C. R. Acad. Sci. Paris 222), with full development in two long 1950 papers in J. Math. Pures Appl. 29. Leray's framework was sheaf-theoretic from the start: given a continuous map $f : E \to B$ , he formed the higher direct image sheaves $R^{q} f_{*} F$ on $B$ and assembled the cohomology of $B$ with these coefficients into a spectral sequence converging to the cohomology of $E$ . The construction was difficult; the differentials were defined recursively, and the multiplicative structure was hard to extract.

Jean-Pierre Serre rebuilt the construction from scratch in his 1951 PhD thesis Homologie singulière des espaces fibrés. Applications (Ann. of Math. 54). Serre's innovation was twofold. First, he used the singular cochain complex of $E$ filtered by the skeletal filtration of $B$ , replacing Leray's sheaf-theoretic machinery with explicit cellular methods that were immediately computable. Second, he made the multiplicative structure transparent by working with cup products on cochains throughout. The thesis used the resulting spectral sequence to compute $π_{4} (S^{3}) = Z /2$ and $π_{n + 1} (S^{n}) = Z /2$ for $n \geq 3$ , the first systematic calculation of low-degree homotopy groups of spheres beyond $π_{n} (S^{n}) = Z$ . Serre's thesis is one of the most consequential single mathematical works of the twentieth century: in 200 pages, he gave the modern formulation of the spectral sequence of a fibration, computed the rational homotopy groups of spheres, proved the finiteness of higher homotopy groups, and laid out the framework that Borel, Cartan, and Adams would build on for the next two decades. He was 25 years old.

The Gysin sequence predates the spectral sequence by a decade. Werner Gysin, a Swiss topologist, established the long exact sequence of an oriented sphere bundle in his 1942 paper Zur Homologietheorie der Abbildungen und Faserungen von Mannigfaltigkeiten (Comment. Math. Helv. 14), in the framework of Eilenberg-Steenrod axiomatic homology. Gysin's original derivation was via the cellular structure of the sphere bundle: the total space deformation-retracts to the base after collapsing each fibre, and the resulting Mayer-Vietoris-type long exact sequence gives the Gysin sequence directly. The connecting map was not initially identified with the Euler class — that identification came later, with Steenrod's 1951 The Topology of Fibre Bundles and the systematic theory of characteristic classes. From the Leray-Serre vantage, the Gysin sequence is the degenerate-spectral-sequence shadow of the more general construction.

Armand Borel's 1953 thesis Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts (Ann. of Math. 57) used the Serre spectral sequence on the fibration $G / T \to B T \to B G$ to compute the cohomology of classifying spaces of compact Lie groups in terms of Weyl-group invariants. This was the gateway to the modern theory of characteristic classes: $H^{*} (B G; Q)$ as the invariants of the Weyl-group action on $H^{*} (B T; Q) = Q [x_{1}, \dots, x_{n}]$ . The Chern, Pontryagin, and Stiefel-Whitney classes acquired their now-standard cohomological characterisations through this framework.

The pedagogical reframing Bott and Tu offer in §14 of Differential Forms in Algebraic Topology (1982) makes the Leray-Serre spectral sequence accessible to a graduate student who has worked through differential forms but not yet seen homological algebra. They construct the spectral sequence directly from the de Rham double complex on a fibre bundle: the rows are the de Rham complex of the fibre, the columns are the fibre integration, and the two filtrations produce the two collapsing routes. This eliminates the singular-chain machinery of Serre's thesis in favour of a directly computable construction in differential forms. The Hopf fibration computation, the Gysin sequence, the Wang sequence, and the Borel computation of $H^{*} (B U (n))$ are all worked through in §14–§17 with the differential-form formalism.

Bibliography [Master]

Leray, J., "L'homologie d'un espace fibré dont la fibre est connexe", J. Math. Pures Appl. 29 (1950), 169–213.
Serre, J.-P., "Homologie singulière des espaces fibrés. Applications", Ann. of Math. 54 (1951), 425–505.
Gysin, W., "Zur Homologietheorie der Abbildungen und Faserungen von Mannigfaltigkeiten", Comment. Math. Helv. 14 (1942), 61–122.
Borel, A., "Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts", Ann. of Math. 57 (1953), 115–207.
Borel, A., "Topology of Lie groups and characteristic classes", Bull. Amer. Math. Soc. 61 (1955), 397–432.
Bott, R. & Tu, L. W., Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer, 1982. §14–§17.
McCleary, J., A User's Guide to Spectral Sequences, 2nd ed., Cambridge University Press, 2001. §5–§6.
Hatcher, A., Spectral Sequences, online supplement to Algebraic Topology, Cornell, 2004.
Steenrod, N., The Topology of Fibre Bundles, Princeton University Press, 1951.

Bott-Tu Pass 4 — Agent C — N8. Leray-Serre spectral sequence with $E_{2} = H^{p} (B; H^{q} (F))$ , multiplicative structure, edge homomorphisms, transgression, Gysin sequence as degenerate Leray-Serre on a sphere bundle. Three Bott-Tu canonical computations: Hopf fibration, $T^{2} \to S^{1}$ collapse, Gysin on $C P^{k}$ . Master Historical channels Serre 1951 PhD thesis directly.

Prerequisites

03.13.01 pending
03.04.07 pending
03.04.13 pending
03.05.02 pending

Used in

03.13.03

Tier anchors

beginner: Bott-Tu §15: cohomology of a fibred space, page-by-page
intermediate: Bott-Tu §15–§17; McCleary §5–§6
master: Serre 1951 PhD thesis; Bott-Tu §14–§17; McCleary §5–§6; Borel *Topology of Lie groups*

References

TODO_REF
Bott-Tu — Differential Forms in Algebraic Topology · §14 'spectral sequence of a fiber bundle'; §15–§17 (Leray-Serre, Gysin, Wang); §20 (Euler class via spectral sequence)
TODO_REF
Serre — Homologie singulière des espaces fibrés. Applications · Ann. of Math. 54 (1951), 425–505 — Serre's PhD thesis
TODO_REF
Leray — L'homologie d'un espace fibré dont la fibre est connexe · J. Math. Pures Appl. 29 (1950), 169–213
TODO_REF
Gysin — Zur Homologietheorie der Abbildungen und Faserungen von Mannigfaltigkeiten · Comment. Math. Helv. 14 (1942), 61–122
TODO_REF
McCleary — A User's Guide to Spectral Sequences · §5 (Leray-Serre), §6 (multiplicative structure and applications)
TODO_REF
Borel — Topology of Lie groups and characteristic classes · Bull. Amer. Math. Soc. 61 (1955), 397–432
TODO_REF
Hatcher — Spectral Sequences (online supplement) · §1.2 (Leray-Serre), §1.4 (Gysin)

Lean module

Codex.SpectralSequences.LeraySerre

Reviewer

TBD

Estimated time

beginner: 22m
intermediate: 55m
master: 80m