Serre spectral sequence

The Serre spectral sequence is a first-quadrant spectral sequence in algebraic topology that arises from a Serre fibration F \to X \to B, where it relates the homology groups H_*(X; G) (or cohomology groups H^*(X; G)) of the total space X to the homology (or cohomology) of the fiber F and base B, with the E_2 page given by E_2^{p,q} = H_p(B; H_q(F; G)) (or E_2^{p,q} = H^p(B; H^q(F; G))) and converging to a filtration of H_*(X; G) (or H^*(X; G)) under suitable conditions such as the base B being simply connected or the fundamental group \pi_1(B) acting trivially on the homology of the fiber.^[1] It was introduced by Jean-Pierre Serre in his 1951 paper "Homologie singulière des espaces fibrés," where he developed it to compute the singular homology of fiber spaces, building on earlier work in sheaf cohomology by Jean Leray. A cohomological version, particularly modulo 2, was further elaborated in Serre's 1953 paper "Cohomologie modulo 2 des complexes d'Eilenberg-MacLane."^[2] This spectral sequence is foundational in homological algebra and topology, providing a systematic way to approximate and compute the homology or cohomology of complex spaces by iteratively resolving differentials on successive pages E_r, which stabilize at E_\infty to yield graded pieces of the associated filtration on the target groups.^[3] It applies to a wide class of fibrations, including those with coefficients in a field or abelian group G, and is particularly powerful when the base or fiber has known homology, allowing deductions about the total space or homotopy groups via the Hurewicz theorem.^[1] Key applications include determining the cohomology of Eilenberg-MacLane spaces K(\mathbb{Z}/2\mathbb{Z}, n), analyzing loop spaces, and computing stable homotopy groups of spheres through Postnikov towers.^[1] The sequence's differentials d_r: E_r^{p,q} \to E_r^{p-r, q+r-1} (in homology) encode higher-order obstructions, making it indispensable for problems where direct computation is infeasible.^[4]

Background and Setup

Serre fibrations

A Serre fibration is a continuous map p: E \to B between topological spaces that satisfies the homotopy lifting property with respect to all CW-complexes. Specifically, for any CW-complex X, any map f: X \to E, and any homotopy H: X \times I \to B starting at p \circ f (where I = [0,1]), there exists a homotopy \tilde{H}: X \times I \to E such that \tilde{H}(x,0) = f(x) and p \circ \tilde{H} = H. This property ensures that the fibration behaves well under homotopy, allowing lifts of homotopies from the base space B to the total space E. Equivalently, the lifting property can be checked just for disks D^k and their boundaries S^{k-1} for all k \geq 0, due to the cellular approximation theorem.^[5] Examples of Serre fibrations include fiber bundles, where the total space E is locally trivial over the base B with fiber F, such as the Hopf bundle S^1 \to S^3 \to S^2. Principal bundles, which are fiber bundles with structure group acting freely and transitively on the fiber, also satisfy the Serre fibration condition. Another prominent example is the path space fibration P B \to B, defined by evaluation at the endpoint, where P B is the space of all paths in B starting from a fixed basepoint, and the fiber over a point is the loop space \Omega B. These examples illustrate how Serre fibrations generalize classical fiber bundles to a broader class suitable for homotopy-theoretic applications.^[5] For a Serre fibration F \to E \xrightarrow{p} B with fiber F = p^{-1}(b_0) over a basepoint b_0 \in B, there exists a long exact sequence of homotopy groups

\cdots \to \pi_n(F) \to \pi_n(E) \to \pi_n(B) \to \pi_{n-1}(F) \to \cdots \to \pi_0(B),

which relates the homotopy groups of the total space, base, and fiber. This sequence implies homotopy equivalences under special conditions, such as when the fiber F is contractible, yielding E \simeq B, or when the base B is contractible, yielding E \simeq F. The exactness captures the interactions between these spaces, providing a tool to compute homotopy groups inductively.^[5] The concept of Serre fibrations was introduced by Jean-Pierre Serre in his 1951 thesis to extend the study of fiber bundles beyond strict local triviality, enabling cohomology computations via a relaxed homotopy lifting condition that suffices for singular homology and cohomology theories. This generalization proved essential for developing spectral sequences in algebraic topology, allowing applications to a wider array of spaces without requiring full bundle structure.

Motivation for the spectral sequence

In the setting of a Serre fibration F \to E \to B, the long exact sequence of homotopy groups relates \pi_*(E), \pi_*(B), and \pi_*(F), but computing the cohomology groups H^*(E) directly is complicated by potential twisting effects from the action of the fundamental group \pi_1(B) on the cohomology of the fiber H^*(F).^[1]^[6] To address these limitations, a filtration is imposed on the cohomology of the total space E based on the skeleta of the base space B, assuming B admits a CW structure. This skeletal filtration generates a sequence of successive approximations to H^*(E), culminating in a spectral sequence that refines the relations between H^*(B) and H^*(F) by incorporating higher-order interactions.^[1]^[6] Jean-Pierre Serre developed this spectral sequence in the early 1950s with the primary aim of computing homotopy groups of spheres, leveraging fibrations such as the Hopf fibration S^1 \to S^3 \to S^2 and relating them to the cohomology of loop spaces via the Hurewicz isomorphism.^[7]^[1]

General Construction

The filtration on cohomology

In the construction of the Serre spectral sequence for the cohomology of a Serre fibration p: E \to B with fiber F, the base space B is equipped with a CW-complex structure. The filtration on the cohomology groups H^n(E; \mathbb{Z}) is defined using the skeletons of B. Specifically, let B_p denote the p-skeleton of B, and let E_p = p^{-1}(B_p) be the preimage in E, which forms a fibration over B_p with the same fiber F. The p-th term of the filtration is then F^p H^n(E; \mathbb{Z}) = \im \left( H^n(E_p; \mathbb{Z}) \to H^n(E; \mathbb{Z}) \right), the image of the induced map on cohomology under the inclusion E_p \hookrightarrow E.^[8]^[1] This filtration is increasing, satisfying F^p H^n(E; \mathbb{Z}) \subset F^{p+1} H^n(E; \mathbb{Z}) for all p, since E_p \subset E_{p+1} induces compatible maps with images satisfying inclusion. It is exhaustive, meaning \bigcup_p F^p H^n(E; \mathbb{Z}) = H^n(E; \mathbb{Z}), as the union of the skeletons \bigcup_p B_p = B implies \bigcup_p E_p = E. The filtration is Hausdorff under suitable conditions, such as when using the complementary definition F^p H^n(E; \mathbb{Z}) = \ker \left( H^n(E; \mathbb{Z}) \to H^n(E, E_{p-1}; \mathbb{Z}) \right), ensuring \bigcap_p F^p H^n(E; \mathbb{Z}) = 0 for spaces of finite type.^[8]^[1] The associated graded pieces of this filtration lead to the first page of the spectral sequence, with E_1^{p,q} \cong H^q(F; \mathbb{Z}), which arises from the long exact sequence in relative cohomology: the relevant quotient corresponds to H^{p+q}(E_p, E_{p-1}; \mathbb{Z}) \cong H^q(F; \mathbb{Z}), since E_p / E_{p-1} is homotopy equivalent to the fiber F over the p-cells of B. These graded pieces on the E_1 page are thus constant along each diagonal p + q = n.^[8]^[1] The construction with integer coefficients assumes that both the base B and the fiber F are simply connected, ensuring the fibration has no nontrivial local coefficient systems and the cohomology of the fibers remains untwisted. In the general case, when \pi_1(B) acts non-trivially on the homology of F, the terms involve local coefficients. This setup extends to oriented fibrations, where the orientation allows for consistent twisting by line bundles, accommodating cases like sphere bundles without full simply connectedness.^[8]^[1]

Definition of the spectral sequence pages

The Serre spectral sequence arises from a filtered cochain complex (C^*, d) associated to the cohomology of the total space E in a Serre fibration, where the filtration \{F^p C^n\} is exhaustive, bounded below, and preserved by the differential d. The bigrading of the spectral sequence is given by integers p \geq 0 and q, with the total degree n = p + q corresponding to the cohomology degree in H^n(E).^[1] The zeroth page E_0^{p,q} is the associated graded object of the filtration, defined as

E_0^{p,q} = \frac{F^p C^{p+q}}{F^{p+1} C^{p+q}},

equipped with the induced differential d^0: E_0^{p,q} \to E_0^{p, q+1} arising from d, which acts within each fixed filtration level p. The first page is the cohomology of this graded complex with respect to d^0:

E_1^{p,q} = H^q(E_0^{p,*}, d^0),

where the cohomology is computed in the q-direction for each fixed p.^[1] Subsequent pages are defined iteratively via differentials d^r: E_r^{p,q} \to E_r^{p+r, q-r+1} of bidegree (r, 1-r) in the (p,q)-grading, satisfying d^r \circ d^r = 0 and preserving the total degree up to sign. The (r+1)-st page is the cohomology of the r-th page:

E_{r+1}^{p,q} = \frac{\ker(d^r: E_r^{p,q} \to E_r^{p+r, q-r+1})}{\operatorname{im}(d^r: E_r^{p-r, q+r-1} \to E_r^{p,q})}.

This process generates a sequence of pages E_r for r = 0, 1, 2, \dots, each refining the previous via these successively longer differentials.^[1] The infinity page E_\infty^{p,q} is the direct limit as r \to \infty, consisting of the permanent cycles modulo boundaries from all higher differentials, and is isomorphic to the associated graded pieces of the induced filtration on the abutment:

E_\infty^{p,q} \cong \frac{F^p H^{p+q}(E)}{F^{p+1} H^{p+q}(E)}.

Under conditions such as the spectral sequence being first-quadrant (with E_r^{p,q} = 0 for p < 0 or q < 0) and the spaces being of finite type, the spectral sequence converges completely to H^*(E), meaning there is a finite filtration on H^n(E) whose associated graded is \bigoplus_p E_\infty^{p,n-p}; more generally, it may converge in the sense of completions if the filtration is complete.^[1]

Cohomology Spectral Sequence

E² page and its computation

The E² page of the cohomology Serre spectral sequence associated to a Serre fibration F \to E \to B, where B is a path-connected CW-complex, is given by

E_2^{p,q} = H^p(B; \mathcal{H}^q(F; \mathbb{Z})),

with p \geq 0, q \geq 0, where \mathcal{H}^q(F; \mathbb{Z}) denotes the local coefficient system on B whose fiber is the cohomology group H^q(F; \mathbb{Z}), determined by the monodromy action of \pi_1(B) on H^*(F; \mathbb{Z}).^[5] This bigrading positions the spectral sequence in the first quadrant, converging to the cohomology of the total space H^{p+q}(E; \mathbb{Z}).^[3] When the action of \pi_1(B) on H^*(F; \mathbb{Z}) is trivial, which occurs if B is simply connected or the fibration is orientable with constant fiber cohomology, the local system \mathcal{H}^q(F; \mathbb{Z}) reduces to the constant coefficient system H^q(F; \mathbb{Z}), yielding

E_2^{p,q} \cong H^p(B; H^q(F; \mathbb{Z})),

which by the universal coefficient theorem is \Hom(H_p(B; \mathbb{Z}), H^q(F; \mathbb{Z})) \oplus \Ext(H_{p-1}(B; \mathbb{Z}), H^q(F; \mathbb{Z})). When working over a field k, this simplifies to H^p(B; k) \otimes H^q(F; k).^[5]^[3] The computation of the E² page proceeds from the skeletal filtration of E induced by the CW-structure of B: let B_p denote the p-skeleton of B and X_p = \pi^{-1}(B_p) the preimage in E. The filtration on the cochain complex C^*(E; \mathbb{Z}) is given by F^p C^n(E; \mathbb{Z}) = C^n(X_p, X_{p-1}; \mathbb{Z}) for n = p+q, yielding an associated graded complex whose cohomology produces the E¹ page. Specifically,

E_1^{p,q} \cong H^{p+q}(X_p, X_{p-1}; \mathbb{Z}) \cong C^p(B_p, B_{p-1}; H^q(F; \mathbb{Z})),

where the isomorphism follows from the product structure over cells and the fact that the relative pairs (X_p, X_{p-1}) are homotopy equivalent to F \times (B_p, B_{p-1}) up to lower skeleta.^[3] The differential d^1: E_1^{p,q} \to E_1^{p+1,q} is induced by the connecting homomorphism \delta in the long exact sequence of the pair (X_{p+1}, X_p),

\cdots \to H^{p+q}(X_{p+1}, X_p) \xrightarrow{\delta} H^{p+q-1}(X_p) \to H^{p+q-1}(X_{p+1}) \to \cdots,

twisted by the action on the fiber coefficients; taking cohomology with respect to d^1 yields the E² page as the cohomology of B with local coefficients in H^*(F; \mathbb{Z}).^[5]^[3] In the general case of nontrivial action, H^p(B; \mathcal{H}^q(F; \mathbb{Z})) is computed as the cohomology with local coefficients, equivalent to the sheaf cohomology \check{H}^p(B; \tilde{\mathcal{H}}^q), where \tilde{\mathcal{H}}^q is the sheafification of the local system \mathcal{H}^q. This accounts for the twisting via deck transformations in the universal cover of B.^[5] The original construction of this spectral sequence, including the role of local systems in the E² term, appears in Serre's foundational work on singular homology of fibrations.

Differentials and convergence

In the cohomology Serre spectral sequence associated to a Serre fibration F \to E \to B with coefficients in an abelian group G, the differentials on the E^r-page are maps d^r: E^r_{p,q} \to E^r_{p+r,q-r+1} of bidegree (r, 1-r). These differentials satisfy d^r \circ d^r = 0, and the E^{r+1}-page is the homology of the E^r-page with respect to d^r. In the first-quadrant spectral sequence arising from a finite filtration, the differentials vanish on E^r_{p,q} for r > q+1, since targets would lie outside the quadrant.^[1] The higher differentials d^r for r \geq 2 are often computed using transgressions or induced by edge effects from the filtration. The transgression is a key map \tau: E^r_{0,q} \to E^r_{r, q-r+1} that arises from the connecting homomorphism in the long exact sequence of the fibration, particularly identifying d^r on the left column E^r_{0,q} with the transgression into the base row E^r_{r, q-r+1}. For instance, elements in E^2_{0,q} \cong H^q(F; G) that survive to E^r_{0,q} may transgress to kill or be killed in the base cohomology.^[1] Under the assumption that the cohomology groups H^*(E; G), H^*(B; G), and H^*(F; G) are finitely generated, the spectral sequence converges strongly to the graded group \mathrm{gr}\, H^n(E; G) = \bigoplus_{p+q=n} E^\infty_{p,q}, meaning there is a finite filtration on H^n(E; G) whose associated graded is the direct sum of the E^\infty-terms. In cases where these groups are not finitely generated, the spectral sequence converges to the completion \widehat{H}^*(E; G) with respect to the filtration.^[1] The edge homomorphisms provide maps from the E^2-page to the abutment: one from E^2_{p,0} \cong H^p(B; G) to H^p(E; G), induced by the projection E \to B, and another from H^{p+q}(E; G) to E^2_{0,q} \cong H^q(F; G), arising from the inclusion of constant paths or sections when available; these are detailed further in the study of boundary maps.^[1] For oriented fibrations with coefficients in a field \mathbb{K}, the Serre spectral sequence is multiplicative, meaning it is a spectral sequence of rings where the E^r-pages inherit a product structure from the cup product in cohomology, E^r_{p,q} \otimes E^r_{s,t} \to E^r_{p+s,q+t}, compatible with the differentials as derivations. This structure facilitates computations involving ring homomorphisms and Steenrod operations.^[1]

Homology Spectral Sequence

E² page for homology

The homology version of the Serre spectral sequence arises from a filtration on the total space E of a Serre fibration F \to E \to B, where B is a CW-complex. The filtration is defined by X_p = \pi^{-1}(B_p), with B_p denoting the p-skeleton of B, inducing a decreasing filtration on the singular chain complex C_*(E) via F_p C_n(E) = \image(C_n(X_p) \to C_n(E)). This setup is dual to the filtration used in the cohomology spectral sequence, shifting focus from cochains to chains while preserving the skeletal structure of the base.^[1] The E^1 page consists of the associated graded pieces of this filtration, given by E^1_{p,q} = H_{p+q}(X_p, X_{p-1}). For fibrations satisfying the Serre conditions (e.g., F and B path-connected, E simply connected if necessary), these relative homology groups are isomorphic to the cellular chains of the base tensored with the homology of the fiber: E^1_{p,q} \cong C_p(B) \otimes H_q(F; \mathbb{Z}), where C_p(B) are the cellular chains of B.^[1]^[6] The first differential d^1: E^1_{p,q} \to E^1_{p-1,q} is induced by the boundary operator in the cellular chain complex of B, tensored with the identity map on H_q(F; \mathbb{Z}); this is denoted \partial \otimes \id and plays an analogous role to the connecting homomorphism in the long exact sequence of the pair (X_p, X_{p-1}), effectively capturing how the fiber homology varies over the base skeleton.^[1]^[6] The E^2 page is the homology of the E^1 page with respect to d^1. In the untwisted case, where \pi_1(B) acts trivially on H_*(F; \mathbb{Z}), this yields

E^2_{p,q} \cong H_p(B; H_q(F; \mathbb{Z})),

the ordinary homology of the base with coefficients in the homology of the fiber as a constant module.^[1] When the action of \pi_1(B) on H_*(F) is nontrivial, the coefficients form a local system \mathcal{H}_q(F) over B, and the E^2 term becomes the homology of the base with local coefficients:

E^2_{p,q} \cong H_p(B; \mathcal{H}_q(F)),

where \mathcal{H}_q(F) is the sheaf of local systems determined by the monodromy action on the fiber's homology. This adjustment accounts for twisting in the fibration without altering the overall convergence to H_{p+q}(E; \mathbb{Z}).^[1]^[6] In algebraic formulations viewing the chain complex of E as derived from a double complex involving C_*(B) and C_*(F), the E^2 page can alternatively be expressed as

E^2_{p,q} \cong \Tor_p^{C_*(B)}(\mathbb{Z}, H_q(F; \mathbb{Z})),

where the Tor is computed over the cellular chains of B; however, the primary computational tool remains the direct evaluation via homology with (local) coefficients.^[6]

Relation to cohomology version

The homology version of the Serre spectral sequence for a fibration F \to E \to B abuts to the homology groups H_*(E; G) of the total space, with the E_2-page given by E_2^{p,q} = H_p(B; \mathcal{H}_q(F; G)), where \mathcal{H}_q(F; G) denotes the local system of coefficients induced by the action of \pi_1(B) on H_q(F; G).^[1] In contrast, the cohomology version abuts to H^*(E; G), with E_2^{p,q} = H^p(B; \mathcal{H}^q(F; G)), and features differentials of opposite degree compared to the homology case.^[1] This duality between the two spectral sequences becomes particularly evident when E, B, and F are closed oriented manifolds, where Poincaré duality induces an isomorphism H_p(B; \mathcal{H}_q(F)) \cong H^{n-p}(B; \mathcal{H}^{m-q}(F)) for dimensions m = \dim F and n = \dim B, assuming compatible orientations.^[1] A key difference arises in the treatment of coefficients: the homology spectral sequence often employs field coefficients, such as \mathbb{Z}/2 or \mathbb{Q}, to ensure direct sum decompositions and avoid extension problems in the filtration, simplifying convergence to H_*(E; G).^[1] The action of \pi_1(B) on the homology groups H_*(F; G) introduces local coefficients similarly to the cohomology case, but computations in homology tend to be more straightforward with fields due to the absence of cup product structures that aid differential calculations in cohomology.^[1] The universal coefficient theorem relates the E_2-pages of the two versions when using integer coefficients and trivial local systems: the homology H_p(B; H_q(F; \mathbb{Z})) can be computed from the cohomology H^*(B; H^q(F; \mathbb{Z})) by first relating H^q(F; \mathbb{Z}) \cong \Hom(H_q(F; \mathbb{Z}), \mathbb{Z}) \oplus \Ext(H_{q-1}(F; \mathbb{Z}), \mathbb{Z}) via UCT on the fiber, and then applying analogous relations on the base, though torsion complicates direct isomorphisms. This highlights how torsion can cause discrepancies between the homology and cohomology pages. The two spectral sequences coincide up to isomorphism when using coefficients in a field like \mathbb{Q}, where homology and cohomology are naturally dual vector spaces without torsion complications, or for free abelian coefficients where the groups are torsion-free and the Ext terms vanish. In such cases, the E_2-pages are isomorphic via the natural duality, and the spectral sequences behave analogously, facilitating computations that leverage results from one version to inform the other.^[1]

Key Properties

Edge and boundary homomorphisms

In the cohomology Serre spectral sequence associated to a Serre fibration F \to E \to B with coefficients in an abelian group G, where B is path-connected and the action of \pi_1(B) on H^*(F; G) is trivial, the edge homomorphisms provide natural maps relating the E_2 page to the cohomology of the total space E. The first edge homomorphism \alpha^p: E_2^{p,0} \to H^p(E; G) is induced by the projection \pi: E \to B, and since E_2^{p,0} = H^p(B; H^0(F; G)) \cong H^p(B; G), it coincides with the induced map \pi^*: H^p(B; G) \to H^p(E; G). The second edge homomorphism \beta^q: H^q(E; G) \to E_2^{0,q} is induced by the inclusion of a fiber j: F \hookrightarrow E, yielding \beta^q = j^*: H^q(E; G) \to H^q(F; G) \cong E_2^{0,q}, assuming H^0(B; H^q(F; G)) \cong H^q(F; G). These maps factor through the associated graded E_\infty^{p,0} and E_\infty^{0,q}, respectively, so \alpha^p induces a monomorphism E_\infty^{p,0} \hookrightarrow H^p(E; G) onto its image, while \beta^q induces an epimorphism from the quotient.^[1]^[9] If all differentials d_r with r \geq 2 vanish on the bottom edge (i.e., d_r^{p,0} = 0 and no incoming differentials to the p-axis), then E_\infty^{p,0} = E_2^{p,0}, making \alpha^p an isomorphism onto its image in H^p(E; G). Similarly, vanishing differentials on the vertical edge (d_r^{0,q} = 0 and no outgoing from the q-axis) imply E_\infty^{0,q} = E_2^{0,q}, so \beta^q is an isomorphism from the kernel of the map to the cokernel. These conditions often arise when the fiber or base has trivial cohomology in certain degrees, allowing the edge maps to detect injections or surjections in the long exact sequence of the fibration.^[1]^[4] The boundary homomorphisms in this context arise from the long exact sequence in cohomology for the fibration, given by

\cdots \to H^n(E; G) \xrightarrow{j^*} H^n(F; G) \xrightarrow{\delta} H^{n+1}(B; G) \xrightarrow{\pi^*} H^{n+1}(E; G) \to \cdots,

where \delta: H^n(F; G) \to H^{n+1}(B; G) is the connecting homomorphism. In the spectral sequence, \delta corresponds to the transgression, realized as the differential d_{n+1}: E_2^{0,n} \to E_2^{n+1,0} when the element survives to that page, mapping a class in H^n(F; G) to a coset in H^{n+1}(B; G) modulo lower filtration terms. This transgression links the vertical edge to the horizontal edge and is central to computing how cohomology classes in the fiber "transgress" to the base.^[1] In the dual homology Serre spectral sequence, with E_2^{p,q} = H_p(B; H_q(F; \pi)) \Rightarrow H_{p+q}(E; \pi), the edge homomorphisms are the duals of the cohomology versions. The base edge map is the composition H_p(E; \pi) \twoheadrightarrow E_\infty^{p,0} \hookrightarrow E_2^{p,0} = H_p(B; \pi), induced by \pi_*: H_p(E; \pi) \to H_p(B; \pi), providing a surjection onto the image. The fiber edge map is E_2^{0,q} = H_q(F; \pi) \twoheadrightarrow E_\infty^{0,q} \hookrightarrow H_q(E; \pi), induced by the inclusion i_*: H_q(F; \pi) \to H_q(E; \pi). Vanishing of relevant differentials again implies these are isomorphisms onto images or from kernels, mirroring the cohomology case. The corresponding boundary homomorphism from the homology long exact sequence,

\cdots \to H_{n+1}(B; \pi) \xrightarrow{\partial} H_n(F; \pi) \to H_n(E; \pi) \to H_n(B; \pi) \to \cdots,

with \partial: H_{n+1}(B; \pi) \to H_n(F; \pi), manifests as the transgression differential d_{n+1}: E_2^{n+1,0} \to E_2^{0,n} in the spectral sequence.^[9]^[4]

Universal coefficient theorem applications

The universal coefficient theorem provides a decomposition of cohomology groups in terms of homology groups with integer coefficients. Specifically, for a space X, it states that

H^n(X; \mathbb{Z}) \cong \Hom(H_n(X; \mathbb{Z}), \mathbb{Z}) \oplus \Ext(H_{n-1}(X; \mathbb{Z}), \mathbb{Z}).

This isomorphism applies directly to the E^2 terms of the Serre spectral sequence in cohomology, where E^2_{p,q} = H^p(B; \mathcal{H}^q(F; \mathbb{Z})) and \mathcal{H}^q(F; \mathbb{Z}) denotes the local coefficient system given by the cohomology of the fiber F with \mathbb{Z}-coefficients.^[1] When the base space B is simply connected, the local system \mathcal{H}^q(F; \mathbb{Z}) is constant, allowing the universal coefficient theorem to decompose each stalk as \mathcal{H}^q(F; \mathbb{Z}) \cong \Hom(H_q(F; \mathbb{Z}), \mathbb{Z}) \oplus \Ext(H_{q-1}(F; \mathbb{Z}), \mathbb{Z}). The E^2 page then splits into two spectral sequences: one with coefficients in the free part \Hom(H_q(F; \mathbb{Z}), \mathbb{Z}) and another in the torsion part \Ext(H_{q-1}(F; \mathbb{Z}), \mathbb{Z}), facilitating the identification of free and torsion components in the cohomology of the total space.^[10]^[1] In the presence of nontrivial local systems arising from a non-simply connected base, the universal coefficient theorem requires additional structure, such as projective or injective resolutions of the coefficient modules over the group ring \mathbb{Z}[\pi_1(B)], to handle the twists induced by the monodromy action. This leads to universal coefficient spectral sequences that resolve the cohomology with local coefficients, where the E^2 page involves \Ext or \Tor terms over the twisted chain complex.^[11]^[1] A key application arises in computations with coefficients in \mathbb{Z}/p\mathbb{Z}, where the universal coefficient theorem yields a natural duality between the homology and cohomology Serre spectral sequences over the field \mathbb{F}_p, with the cohomology version being the vector space dual of the homology version. This reduction simplifies calculations by allowing the use of field coefficients, avoiding extensions and torsion issues in the integer case, and often collapses the sequence to directly reveal the mod-p structure of the total space's homology or cohomology.^[10]^[1]

Example Computations

Hopf fibration

The classical Hopf fibration is the fiber bundle S^1 \to S^3 \to S^2, where the total space is the 3-sphere, the base is the 2-sphere, and the fiber is the circle. This fibration provides a fundamental example for applying the Serre spectral sequence in cohomology with integer coefficients. The cohomology ring of the base is H^*(S^2; \mathbb{Z}) \cong \mathbb{Z} / (x^2), where x is a generator in degree 2. The cohomology of the fiber is the exterior algebra H^*(S^1; \mathbb{Z}) \cong \Lambda(y), where y generates in degree 1.^[8]^[1] Since the base S^2 is simply connected, the local coefficient system is trivial, and the E_2 page of the Serre spectral sequence is given by E_2^{p,q} = H^p(S^2; H^q(S^1; \mathbb{Z})) \cong H^p(S^2; \mathbb{Z}) \otimes H^q(S^1; \mathbb{Z}). The nonzero entries are thus \mathbb{Z} at positions (p,q) = (0,0), (0,1), (2,0), and (2,1), corresponding to the tensor products $1 \otimes 1, $1 \otimes y, x \otimes 1, and x \otimes y, respectively. The spectral sequence is multiplicative, so its algebra structure is induced by that of the base and fiber cohomologies.^[8]^[1] The first nontrivial differential is the d_2-transgression, which maps the generator y of E_2^{0,1} isomorphically to the generator x of E_2^{2,0}. This differential respects the multiplicative structure: for the class x \otimes y in E_2^{2,1}, we have d_2(x y) = x \cdot d_2(y) = x \cdot x = x^2 = 0, since x^2 = 0 in H^*(S^2; \mathbb{Z}). Thus, d_2 kills the terms in (0,1) and (2,0), while the term in (2,1) survives to E_3 = E_\infty. Higher differentials vanish on the remaining terms, as their targets lie outside the page.^[8]^[1] The spectral sequence converges to the cohomology of the total space: E_\infty^{0,0} \cong \mathbb{Z} contributes to H^0(S^3; \mathbb{Z}) \cong \mathbb{Z}, and E_\infty^{2,1} \cong \mathbb{Z} contributes to H^3(S^3; \mathbb{Z}) \cong \mathbb{Z}, with all other groups zero. The filtration on H^3(S^3; \mathbb{Z}) has F^2 H^3 / F^3 \cong E_\infty^{2,1} \cong \mathbb{Z} and F^3 H^3 = 0, confirming the isomorphism. This computation demonstrates how the nontrivial fibration structure is encoded in the transgression, which obstructs direct Künneth isomorphism and reveals the generator of H^3(S^3) as arising from the product x y in the E_2 page.^[8]^[1]

Path space fibration

The path space fibration is given by the evaluation map \mathrm{ev}: P(X) \to X, where P(X) denotes the space of continuous paths \gamma: [0,1] \to X with \gamma(0) = x_0 for a fixed basepoint x_0 \in X, and \mathrm{ev}(\gamma) = \gamma(1). The fiber over the basepoint x_0 is the loop space \Omega X, consisting of loops based at x_0. This fibration is a Serre fibration, and the total space P(X) is contractible, as it admits a homotopy to the constant path at x_0 via linear reparametrization.^[1] Applying the Serre spectral sequence in cohomology with integer coefficients (assuming X is path-connected and simply connected to ensure trivial local coefficients), the E_2-page is E_2^{p,q} = H^p(X; H^q(\Omega X; \mathbb{Z})), which converges to the cohomology of the total space H^{p+q}(P(X); \mathbb{Z}). Since P(X) is contractible, H^k(P(X); \mathbb{Z}) = \mathbb{Z} for k = 0 and $0 otherwise. Thus, the only nonzero E_\infty-term is E_\infty^{0,0} \cong \mathbb{Z}, implying that all other entries on the E_2-page must be killed by higher differentials.^[1]^[10] The first-quadrant structure of the spectral sequence ensures that the p=0 column E_2^{0,q} = H^q(\Omega X; \mathbb{Z}) and the q=0 row E_2^{p,0} = H^p(X; \mathbb{Z}) interact via differentials d_r: E_r^{p,q} \to E_r^{p+r, q-r+1}. These differentials, particularly the transgressions along the edges, induce isomorphisms H^q(\Omega X; \mathbb{Z}) \cong \tilde{H}^{q+1}(X; \mathbb{Z}) for q \geq 1, where \tilde{H}^\bullet denotes reduced cohomology. This relation embodies the Hurewicz isomorphism in cohomology, establishing a direct link between the cohomology of the loop space and that of the base space.^[1]^[10] This application of the Serre spectral sequence to the path space fibration provides a foundational tool for computing the cohomology of loop spaces from the cohomology of the base, facilitating broader homotopy-theoretic investigations such as stable homotopy groups or Eilenberg-MacLane space structures. Edge homomorphisms may identify surviving terms on the axes before differentials act.^[1]

Complex projective space cohomology

The Serre spectral sequence applied to the Hopf fibration S^1 \to S^{2n+1} \to \mathbb{CP}^n provides a method to compute the cohomology of complex projective space \mathbb{CP}^n. This fibration arises from the action of S^1 on S^{2n+1} \subset \mathbb{C}^{n+1} by complex multiplication on coordinates, yielding the quotient \mathbb{CP}^n as the base space.^[5] The E_2 page of the cohomology spectral sequence with \mathbb{Z} coefficients is given by

E_2^{p,q} = H^p(\mathbb{CP}^n; H^q(S^1; \mathbb{Z})),

where the local coefficient system is trivial due to the simply connectedness of the base. The cohomology of the fiber is H^*(S^1; \mathbb{Z}) = \wedge(\sigma) with |\sigma| = 1, so H^q(S^1; \mathbb{Z}) = \mathbb{Z} for q = 0, 1 and 0 otherwise. Thus, the E_2 page consists of two rows isomorphic to the unknown H^*(\mathbb{CP}^n; \mathbb{Z}), one in total degree p (for q=0) and one in total degree p+1 (for q=1). The sequence converges to H^*(S^{2n+1}; \mathbb{Z}), which is \mathbb{Z} in degrees 0 and $2n+1, and 0 elsewhere.^[12]^[13] The first nontrivial differentials are the d_2 maps d_2: E_2^{p,1} \to E_2^{p+2,0}. Due to the multiplicativity of the spectral sequence, these are determined by the transgression d_2(\sigma) = \iota, where \iota generates H^2(\mathbb{CP}^n; \mathbb{Z}). This implies d_2(\iota^k \cdot \sigma) = \iota^{k+1} for k \geq 0. The differentials kill all entries in the q=1 row except the class \iota^n \cdot \sigma in bidegree (2n, 1), whose image d_2(\iota^n \cdot \sigma) = \iota^{n+1} must be zero to allow survival to E^\infty and account for the generator of H^{2n+1}(S^{2n+1}; \mathbb{Z}). No further nontrivial differentials appear, as higher d_r (r \geq 3) would target negative q-degrees from the q=0 row or exceed the filtration bounds.^[12]^[14] The E^\infty page therefore has \mathbb{Z} at bidegrees (0,0) and (2n,1), and zero elsewhere. The chain of isomorphisms given by the d_2 differentials implies that H^{2k}(\mathbb{CP}^n; \mathbb{Z}) \cong \mathbb{Z} for k = 0, \dots, n, and zero in odd degrees and higher even degrees. Since the filtration is by even degrees and the groups are free, H^*(\mathbb{CP}^n; \mathbb{Z}) is \mathbb{Z} in even degrees 0 through $2n and 0 in odd degrees. The multiplicativity of the spectral sequence ensures the ring structure on the edge is preserved, yielding the cohomology ring

H^*(\mathbb{CP}^n; \mathbb{Z}) \cong \mathbb{Z}[\iota]/(\iota^{n+1}),

where |\iota| = 2. This computation extends inductively from lower-dimensional projective spaces, as the relation \iota^{n+1} = 0 truncates the polynomial algebra built from the generator in degree 2.^[5]^[13]

Homotopy groups of spheres

The Serre spectral sequence provides a powerful tool for computing homotopy groups of spheres through its application to fibrations arising in the Postnikov tower of S^n. For suitable fibrations arising in Postnikov towers of simply connected spaces, the Serre cohomology spectral sequence, combined with the Hurewicz theorem, relates the cohomology of intermediate spaces to homotopy groups of the fibers, allowing computations of homotopy groups such as those of spheres. This approach allows the homotopy groups of the total space E to be derived from the cohomology of the base and the homology of the fiber, with differentials encoding higher-order obstructions.^[1] Jean-Pierre Serre pioneered this approach in his analysis of spheres, using the spectral sequence to establish the finiteness properties of \pi_*(S^n). In particular, Serre's finiteness theorem states that for n \geq 2, the groups \pi_k(S^n) are finite abelian groups whenever k \neq n and k \neq 2n-1 with n even; in the exceptional case k = 4m-1 and n = 2m even, \pi_{4m-1}(S^{2m}) \cong \mathbb{Z} \oplus F_m where F_m is finite. This result was proved by examining the spectral sequence for the path-loop fibration \Omega S^n \to P S^{n-1} \to S^{n-1} and related constructions, showing that infinite cyclic factors only appear in specific dimensions tied to the Hopf invariant. Specific computations illustrate the method's effectiveness in low dimensions. For instance, to compute \pi_4(S^3), consider the Postnikov tower stage where the fibration is K(\mathbb{Z}_2, 4) \to X_4 \to K(\mathbb{Z}, 3), with S^3 as the homotopy fiber of the next stage; the Serre spectral sequence for the cohomology of X_4 yields E_2^{p,q} with a nontrivial term at E_2^{3,2} \cong \mathbb{Z}/2\mathbb{Z}, and the only surviving E_\infty term on the line p+q=5 is E_\infty^{3,2} \cong \mathbb{Z}/2\mathbb{Z}, implying H^5(X_4; \mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}. By the Hurewicz theorem and universal coefficient theorem, this forces \pi_4(S^3) \cong \mathbb{Z}/2\mathbb{Z}. Similar analyses compute \pi_{n+2}(S^n) \cong \mathbb{Z}/2\mathbb{Z} for n \geq 3, arising from a nontrivial differential related to the Steenrod square Sq^2.^[1] Serre's work also revealed torsion patterns, such as the p-primary component of \pi_{n+2p-3}(S^n) \cong \mathbb{Z}/p\mathbb{Z} for primes p and n \geq 3 in the metastable range n < 2p-2, detected via differentials in the spectral sequence for Eilenberg-MacLane space fibrations. These results highlight the sequence's role in distinguishing infinite and torsion elements, with further refinements using the EHP sequence to relate unstable groups across dimensions. For example, the Hopf fibration S^3 \to S^7 \to \mathbb{C}P^3 contributes to understanding \pi_7(S^4) \cong \mathbb{Z} \oplus \mathbb{Z}/12\mathbb{Z}, where the integer factor stems from the Hopf invariant one, and torsion from spectral sequence extensions. Overall, the Serre spectral sequence not only computes explicit groups but also structures the stable and unstable homotopy of spheres, influencing subsequent developments in Adams spectral sequences for higher stems.^[1]