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Fibration

In algebraic topology, a fibration, specifically a Serre fibration, is defined as a continuous surjective map p: E \to B between topological spaces E (the total space) and B (the base space) that satisfies the homotopy lifting property with respect to all CW-complexes.^[1] This property ensures that for any map f: X \to E and homotopy H: X \times I \to B with p \circ f = H(-,0), there exists a lift \tilde{H}: X \times I \to E such that p \circ \tilde{H} = H and \tilde{H}(-,0) = f.^[1] Fibrations generalize fiber bundles, which are locally trivial fibrations where the preimage p^{-1}(U) over open sets U \subset B is homeomorphic to U \times F for a fixed fiber F, but relax the strict local triviality to a homotopical condition.^[2] The concept of fibrations evolved from early work on fiber spaces in the 1930s, with Herbert Seifert introducing fiber spaces in 1932 as a tool for studying three-dimensional manifolds and the Poincaré conjecture through decompositions into fibrations with circle fibers.^[3] Witold Hurewicz advanced the theory in 1935 by defining higher homotopy groups, and in 1941 with Norman Steenrod established a long exact sequence relating the homotopy groups of the total space, base, and fiber in such structures.^[3] Jean-Pierre Serre formalized the modern notion of Serre fibrations in 1951, emphasizing the covering homotopy property (a special case of homotopy lifting for disks and spheres) to facilitate homological and cohomological computations in homotopy theory.^[1] By the early 1950s, fibrations had become central to algebraic topology, as detailed in Norman Steenrod's 1951 textbook The Topology of Fibre Bundles and international conferences like those in Bruxelles (1950) and Cornell (1953).^[3] Fibrations are essential for analyzing topological spaces via fiber sequences, yielding long exact sequences in homotopy groups \cdots \to \pi_n(F) \to \pi_n(E) \to \pi_n(B) \to \pi_{n-1}(F) \to \cdots, where F is the fiber, which underpin computations like those for homotopy groups of spheres.^[2] They also enable powerful tools such as the Serre spectral sequence, which converges to the homology or cohomology of E from those of B and F, with E_2^{p,q} = H_p(B; H_q(F)).^[2] In Postnikov towers, fibrations decompose simply connected spaces into layers modeled by Eilenberg-MacLane spaces, capturing higher homotopy groups via cohomology classes.^[1] Beyond topology, the notion extends to category theory as Grothendieck fibrations, where a functor p: \mathcal{E} \to \mathcal{B} between categories satisfies a Cartesian lifting property, modeling dependent types and higher parallel transport in homotopy type theory.^[4] Key examples include covering spaces (discrete fibrations), principal G-bundles classified by maps to the classifying space BG, and the path space fibration PX \to X with fiber the loop space \Omega X.^[2]

Formal Definitions

Homotopy Lifting Property

The homotopy lifting property (HLP) is a key axiom in algebraic topology for a continuous map p: E \to B between topological spaces. It asserts that for every topological space X, every continuous map f: X \to E, and every continuous homotopy H: X \times I \to B with H(x, 0) = p(f(x)) for all x \in X, there exists a continuous homotopy \tilde{H}: X \times I \to E such that \tilde{H}(x, 0) = f(x) and p \circ \tilde{H} = H.^[5]^[6] This condition is captured by a commutative diagram, where the solid arrows denote the given maps and homotopies, and the dashed arrow indicates the required lift:

\begin{CD} X @>f>> E \\ @V{i_0}VV @VVpV \\ X \times I @>H>> B \end{CD}

with i_0: X \to X \times I the inclusion at time 0, and the lift \tilde{H} making the right triangle commute while matching f initially.^[5] The HLP is essential for maps like fibrations because it guarantees that homotopies in the base space B can be lifted to the total space E, thereby preserving essential homotopy-theoretic structures of fiber bundles.^[5] This lifting capability underpins the derivation of long exact sequences in homotopy groups for such maps and facilitates advanced tools like spectral sequences in homology.^[5] The property was introduced by Jean-Pierre Serre in 1951 as a homotopy analogue to exact sequences, enabling the systematic computation of singular homology for fibrations. Maps satisfying the HLP with respect to all CW-complexes X are termed Serre fibrations.^[5]

Serre Fibrations

A Serre fibration is a continuous surjective map p: E \to B between topological spaces that possesses the homotopy lifting property with respect to all CW-complexes. This property ensures that for any CW-complex A, any continuous map f: A \to E, and any homotopy G: A \times I \to B such that G_0 = p \circ f, there exists a homotopy H: A \times I \to E with H_0 = f and p \circ H = G. Equivalently, since every topological space is weakly homotopy equivalent to a CW-complex, the homotopy lifting property holds with respect to all topological spaces via cellular approximation. The fiber of p over a point b \in B is the preimage p^{-1}(b), a subspace of the total space E; when all fibers are homotopy equivalent, this is referred to as the typical fiber F. A key characterization arises through the singular simplicial set functor: the map p is a Serre fibration if and only if the induced map \mathrm{Sing}(p): \mathrm{Sing}(E) \to \mathrm{Sing}(B) is a Kan fibration, meaning it has the right lifting property against all horn inclusions in the category of simplicial sets. In the category of topological spaces, Serre fibrations serve as the fibrations in Quillen's closed model category structure, where weak equivalences are weak homotopy equivalences and cofibrations are relative cell complexes with the left lifting property against acyclic fibrations. This model structure is cofibrantly generated, and the class of Serre fibrations is closed under pullbacks: if p: E \to B is a Serre fibration and f: B' \to B is any continuous map, then the pulled-back map p' : E' \to B' (where E' = \{ (e, b') \in E \times B' \mid p(e) = f(b') \}) is also a Serre fibration. Unlike classical fiber bundles, which are locally trivial (i.e., admit local trivializations over open covers of the base), Serre fibrations need not be locally trivial but inherit essential homotopy-theoretic features.

Grothendieck Fibrations

A Grothendieck fibration is a functor p: \mathcal{E} \to \mathcal{B} between categories such that for every object e \in \mathcal{E} with b = p(e) and every morphism f: b' \to b in \mathcal{B}, there exists a lift \tilde{f}: e' \to e in \mathcal{E} with p(\tilde{f}) = f, called a Cartesian morphism, satisfying the following universal property: for any other morphism h: e'' \to e in \mathcal{E} with p(h) = g \circ f for some g: b'' \to b' in \mathcal{B}, there exists a unique morphism k: e'' \to e' in \mathcal{E} with p(k) = g such that h = \tilde{f} \circ k.^[7] This ensures that the fibers p^{-1}(b) over objects b \in \mathcal{B} can be reindexed along morphisms in \mathcal{B}, providing a way to transport structure covariantly. A morphism in \mathcal{E} is Cartesian with respect to p if it satisfies the above lifting property universally for any compatible morphism in the codomain. Dually, for opfibrations, one considers cocartesian morphisms, where lifts exist contravariantly: for f: b \to b' in \mathcal{B} and e \in \mathcal{E} over b, there is a cocartesian lift \tilde{f}: e \to e' over f, with the property that any other lift factors uniquely through it.^[7] A bifibration is a functor that is both a fibration and an opfibration, often arising when the reindexing functors f^*: p^{-1}(b') \to p^{-1}(b) admit left adjoints, ensuring balanced transport in both directions.^[7] Named after Alexander Grothendieck, this notion was introduced in his 1961 seminar to formalize descent theory and the study of sheaves in algebraic geometry, where fibrations organize local-to-global constructions over sites.^[8] In this context, Grothendieck fibrations generalize pseudofunctors \mathcal{B}^{op} \to \mathbf{Cat}, allowing the encoding of indexed categories via the Grothendieck construction, which integrates them into a single fibered category.^[7] A key example arises in algebraic geometry over the étale topology on schemes, where the fibered category \mathbf{QCoh}/\mathbf{Sch}/S of quasi-coherent sheaves assigns to each scheme the category of quasi-coherent modules, with pullback functors f^* providing Cartesian lifts along étale morphisms; this structure facilitates descent for sheaves, ensuring they glue effectively over étale covers.^[9] While analogous to Serre fibrations in preserving lifting properties, Grothendieck fibrations operate in the abstract categorical setting to manage reindexing rather than topological homotopy.^[7]

Examples

Trivial Fibrations

In homotopy theory, a trivial fibration is defined as a Serre fibration p: E \to B that is also a weak homotopy equivalence. This means that p induces isomorphisms on all homotopy groups \pi_n(E, e_0) \to \pi_n(B, b_0) for n \geq 0, where e_0 \in E maps to b_0 \in B.^[10] Equivalently, a trivial fibration is a Serre fibration with contractible fibers F = p^{-1}(b), since the long exact sequence of homotopy groups for a Serre fibration

\cdots \to \pi_{n+1}(B, b_0) \to \pi_n(F, f_0) \to \pi_n(E, e_0) \to \pi_n(B, b_0) \to \pi_{n-1}(F, f_0) \to \cdots

then implies that \pi_n(F, f_0) = 0 for all n yields isomorphisms \pi_n(E, e_0) \cong \pi_n(B, b_0), making p a weak equivalence (assuming E and B are nice spaces like CW complexes, where weak equivalences are homotopy equivalences).^[10] Such fibrations are characterized as those liftable to homotopy equivalences in the sense that there exists a homotopy equivalence section s: B \to E such that p \circ s \simeq \mathrm{id}_B, or more generally, as acyclic assemblies where the assembly map is a weak equivalence with contractible kernel. Trivial fibrations can be constructed as product projections E = B \times F \to B where F is any contractible space, such as a point \{*\}, yielding the canonical example \pi_1: B \times \{*\} \to B.^[10] In this case, the homotopy lifting property holds trivially, as lifts can be defined componentwise using contractions of F. More generally, any Serre fibration over a contractible base B is fiber homotopy equivalent to such a product projection.^[10] In the classical (Serre) model structure on the category of topological spaces, where weak equivalences are weak homotopy equivalences, fibrations are Serre fibrations, and cofibrations are retracts of relative cell complexes, the trivial fibrations are precisely the acyclic fibrations—i.e., the fibrations that are weak equivalences. This identification follows from the model category axioms, ensuring that trivial fibrations have the right lifting property with respect to acyclic cofibrations.^[11] Trivial fibrations play a key role in obstruction theory by classifying homotopically split extensions: the existence of a homotopy section to a fibration corresponds to vanishing obstructions in cohomology groups H^{n+1}(B; \pi_n(F)), and when F is contractible, such extensions split homotopically, reducing to products up to homotopy equivalence.^[10]

Hopf Fibration

The Hopf fibration is a canonical example of a non-trivial Serre fibration, defined as the continuous surjective map h: S^3 \to S^2 with fiber S^1, where S^3 is identified with the unit sphere in \mathbb{C}^2 consisting of pairs (z_1, z_2) such that |z_1|^2 + |z_2|^2 = 1, and the map is given explicitly by

h(z_1, z_2) = \left( 2 z_1 \overline{z_2}, |z_1|^2 - |z_2|^2 \right),

with the image in \mathbb{C} \times \mathbb{R} \cong \mathbb{R}^3 lying on S^2.^[12]^[13] This construction arises from viewing S^3 as the unit quaternions and projecting via conjugation by the imaginary unit quaternion i, yielding h(q) = q i \overline{q} for unit quaternion q, which produces the same fibration.^[13] Geometrically, the Hopf fibration realizes S^3 as the unit sphere bundle of the tautological complex line bundle over \mathbb{CP}^1, which is diffeomorphic to S^2; here, points in the total space correspond to lines in \mathbb{C}^2 equipped with a unit vector in that line, and the projection forgets the vector while retaining the line.^[14] The fibers are great circles on S^3, linked in a manner that any two distinct fibers are interlocked with linking number one, illustrating the non-trivial topology.^[12] As a principal U(1)-bundle (equivalently, S^1-bundle), the Hopf fibration is not trivializable over S^2, distinguishing it from product bundles; its classifying map generates the homotopy group \pi_3(S^2) \cong \mathbb{Z}, providing the first example of a non-zero higher homotopy group for spheres.^[12] This non-triviality can be verified using the long exact sequence of homotopy groups for the fibration, which reveals the generator of \pi_3(S^2).^[12] Discovered by Heinz Hopf in his 1931 paper, the fibration provided foundational evidence for the richness of homotopy groups of spheres, influencing subsequent developments in algebraic topology. Higher-dimensional analogs exist as fibrations S^{2n+1} \to \mathbb{CP}^n with fiber S^1 for n > 1, generalizing the construction via unit spheres in \mathbb{C}^{n+1} and projection to projective space.^[12]^[13]

Basic Constructions

Path Space Fibration

In algebraic topology, given a pointed topological space (X, x_0), the path space P(X) consists of all continuous paths \gamma: [0,1] \to X such that \gamma(0) = x_0, equipped with the compact-open topology. The path space fibration is the evaluation map \mathrm{ev}_1: P(X) \to X defined by \mathrm{ev}_1(\gamma) = \gamma(1), whose fiber over the basepoint x_0 is the loop space \Omega X = \{\gamma \in P(X) \mid \gamma(1) = x_0\}.^[5] The space P(X) is contractible, ensuring the fibration captures essential homotopy information through its fiber.^[15] The evaluation map at the initial parameter, \mathrm{ev}_0: P(X) \to X given by \mathrm{ev}_0(\gamma) = \gamma(0), is the constant map to x_0. The inclusion \iota: \Omega X \hookrightarrow P(X) of loops as paths starting and ending at x_0 composes with \mathrm{ev}_1 to yield the constant map to x_0, forming the fibration sequence \Omega X \to P(X) \to X. This sequence exemplifies a principal fibration, serving as a universal model from which other fibrations with loop space fibers can be derived.^[5] The path space fibration is a Serre fibration, satisfying the homotopy lifting property with respect to all topological spaces. Iterating the construction—applying it successively to the loop spaces \Omega^k X for k \geq 1—produces a tower of fibrations whose connectivity relates the higher homotopy groups of X to the path components of the iterated loop spaces via the isomorphism \pi_n(X, x_0) \cong \pi_0(\Omega^n X, \mathrm{const}_{x_0}).^[5]^[15] In the context of model categories, the path space fibration generalizes to the notion of a path space object for fibrant objects, providing a categorical analog that facilitates homotopy relations and resolutions in simplicial homotopy theory. The inclusion of constant loops, specifically the constant path at x_0 serving as the basepoint in \Omega X, acts as a canonical section for the fiber over x_0, enabling the identification of trivial homotopy classes within the loop space.^[16]^[15]

Pullback Fibrations

In algebraic topology, the pullback construction provides a method to induce a new fibration over a different base space from an existing fibration via a base change map. Given a Serre fibration p: E \to B and a continuous map f: B' \to B, the pullback fibration is defined as the projection p': E' \to B', where E' = \{ (b', e) \in B' \times E \mid f(b') = p(e) \} is the pullback of p along f, and p'(b', e) = b'.^[17] This construction forms a Cartesian square:

\begin{CD} E' @>{\pi}>> E \\ @V{p'}VV @VV{p}V \\ B' @>>f> B, \end{CD}

where \pi: E' \to E is the projection \pi(b', e) = e, and the diagram commutes since p \circ \pi = f \circ p'. A fundamental property is that pullbacks preserve the Serre fibration structure: if p: E \to B is a Serre fibration, then so is p': E' \to B'.^[17] Moreover, the fibers are preserved up to isomorphism, with the fiber (f^* p)_{b'} \cong p_{f(b')} for each b' \in B', ensuring that the pullback fibration has the same local structure as the original over corresponding points. In the context of category theory, a similar pullback operation applies to Grothendieck fibrations. For a Grothendieck fibration P: \mathcal{E} \to \mathcal{B} (a functor admitting Cartesian lifts of morphisms in \mathcal{B}) and a functor f: \mathcal{B}' \to \mathcal{B}, the pullback fibration f^* P: f^* \mathcal{E} \to \mathcal{B}' is constructed by forming the fibered category over \mathcal{B}' whose objects and morphisms are pairs consisting of those in \mathcal{B}' and Cartesian lifts in \mathcal{E} along f. This pullback yields another fibered category, preserving the Grothendieck fibration property, as the Cartesian lift condition is stable under base change. The resulting square of categories is Cartesian in the sense that it faithfully represents the reindexing of fibers along f. This construction is particularly useful in homotopy theory for altering base spaces to simplify computations. In model category theory, pullbacks preserve fibrations, meaning that if p is a fibration in a model category, then the pullback along any morphism is also a fibration; this contrasts with pushouts, which instead preserve cofibrations.^[18] The path space fibration serves as a universal example of such a pullback along the evaluation map from the path space.

Principal Fibrations

A principal G-fibration is a Serre fibration p: P \to B with fiber G, where G is a discrete or topological group, G acts freely and properly on P from the right, and p is G-equivariant, meaning p(pg) = p(p) for all p \in P and g \in G.^[2] This structure generalizes principal G-bundles to the homotopy-theoretic setting of Serre fibrations, allowing for non-locally trivial cases while preserving the essential equivariant properties.^[19] From a principal G-fibration P \to B, one constructs associated fibrations by taking a left G-space F (arising from a representation of G on F) and forming the quotient space E = P \times_G F = (P \times F)/G, where (p, f) \sim (pg, g^{-1}f) for g \in G; the projection E \to B is then a fibration with fiber F.^[2] This construction shows how principal G-fibrations classify more general fibrations with structure group G, as any such fibration arises as an associated fibration to a principal one via a choice of frame or section over the fiber.^[20] A key theorem states that every oriented fibration with fiber \mathbb{R}^n is associated to a principal SO(n)-fibration (noting that orientation reduces the structure group from O(n) to SO(n)), enabling the classification of such fibrations through the classifying space BSO(n).^[20] Principal G-fibrations over a base B are classified up to fiber homotopy equivalence by homotopy classes of maps [B, BG], where BG is the classifying space for G; for bundles over simplicial complexes, this reduces to clutching functions in \pi_{n-1}(G) over n-spheres or Čech cohomology H^1(B; \underline{G}) when G is discrete.^[2] In differential geometry, the tangent bundle TM \to M of a Riemannian manifold M is the associated bundle to the orthonormal frame bundle P \to M, which is a principal O(n)-fibration, illustrating how principal fibrations encode the geometric structure of the base.^[20] The concept of principal fibrations was formalized in the 1950s by Norman Steenrod in his development of homotopy theory for fiber bundles, extending classical bundle theory to fibrations.^[21] For example, the Hopf fibration S^3 \to S^2 with fiber S^1 is a principal U(1)-fibration.^[2]

Homotopy Properties

Fiber Homotopy Equivalence

In a fibration p: E \to B, the fibers F_b = p^{-1}(b) and F_{b'} = p^{-1}(b') over base points b, b' \in B are homotopy equivalent if there exists a homotopy equivalence f: F_b \to F_{b'} induced by lifting a path \gamma: I \to B from b to b' via the homotopy lifting property, yielding a fiber-preserving map that is a homotopy equivalence in the homotopy category of spaces.^[1]^[22] For a Serre fibration over a path-connected base B, all fibers are homotopy equivalent, as paths between any base points induce such equivalences, forming a functor from the path components of B to the homotopy category.^[1] If the fibration is orientable—meaning the monodromy action preserves orientation on the fibers—these homotopy equivalences can be chosen coherently across the base, ensuring a consistent orientation on the fiber homotopy types.^[23] In contrast, non-orientable fibrations may yield homotopy equivalences that are not equivariant under the twisting monodromy. A map of Serre fibrations (f, g): (E \to B) \to (E' \to B'), where p' \circ f = g \circ p, is a fiber homotopy equivalence if f is fiber-preserving and a homotopy equivalence on total spaces, thereby inducing homotopy equivalences on corresponding fibers F_b \to F'_{g(b)} and on the base via g.^[1]^[24] This notion extends to abstract homotopy theory, where fiber homotopy equivalences underpin the definition of homotopy fibers in model categories, replacing strict point-set fibers with weakly equivalent approximations to capture essential homotopy data in derived categories.^[1]^[18] A counterexample arises in the projection of the Möbius band to its base circle, a non-orientable line bundle where fibers are intervals (homotopy equivalent as contractible spaces), but the homotopy equivalences induced by path lifting are not equivariant under the monodromy twist, which reverses orientation along non-trivial loops.^[23]

Long Exact Sequence of Homotopy Groups

In algebraic topology, a fundamental consequence of Serre fibrations is the existence of a long exact sequence relating the homotopy groups of the total space, base space, and fiber. For a Serre fibration p: E \to B with fiber F = p^{-1}(b_0) over a basepoint b_0 \in B, assuming B is path-connected, there is a long exact sequence

\cdots \to \pi_{n+1}(B, b_0) \xrightarrow{\partial} \pi_n(F, e_0) \xrightarrow{i_*} \pi_n(E, e_0) \xrightarrow{p_*} \pi_n(B, b_0) \to \pi_{n-1}(F, e_0) \to \cdots,

where e_0 \in F is a basepoint, i: F \to E is the inclusion, and the maps are induced by inclusion, projection, and the boundary homomorphism \partial.^[5] This sequence holds for all n \geq 1, with exactness meaning that at each group, the image of the incoming map equals the kernel of the outgoing map.^[5] The derivation of this sequence follows from the Puppe sequence applied to the fibration, which is the cofiber sequence F \to E \to B \to \Sigma F \to \Sigma E \to \Sigma B, where \Sigma denotes suspension.^[25] The boundary map \partial: \pi_{n+1}(B) \to \pi_n(F) is constructed using the homotopy lifting property of Serre fibrations: given a representative sphere S^{n+1} \to B, lift it to a disk D^{n+1} \to E, and the boundary of this disk in the fiber F defines the image under \partial.^[5] Exactness is then verified using the five-lemma on appropriate short exact sequences induced by the fibration and path-space constructions.^[5] Special cases arise depending on the connectivity of the fiber. If F is (n-1)-connected, meaning \pi_k(F) = 0 for all k \leq n-1, then exactness implies that p_*: \pi_k(E) \to \pi_k(B) is an isomorphism for k < n and surjective for k = n.^[5] For the low-dimensional case n=1, the sequence yields a five-term exact sequence \pi_1(F) \to \pi_1(E) \to \pi_1(B) \to \pi_0(F) \to \pi_0(E) \to \pi_0(B), which terminates since \pi_0(B) = 0 under the path-connectedness assumption on B.^[5] This sequence serves as a key computational tool, as illustrated by the Hopf fibration S^1 \to S^3 \to S^2. Here, the relevant portion is \pi_3(S^1) \xrightarrow{i_*} \pi_3(S^3) \xrightarrow{p_*} \pi_3(S^2) \xrightarrow{\partial} \pi_2(S^1) = 0, and since \pi_3(S^1) = 0 and p_* is an isomorphism \mathbb{Z} \to \mathbb{Z} (the Hopf map itself), exactness implies \pi_3(S^2) \cong \mathbb{Z}, detecting the Hopf invariant one generator.^[5] The result extends to more abstract settings: for Grothendieck fibrations in homotopy categories, exactness of the induced sequence on homotopy groups holds via derived functors of the fiber functor, preserving the boundary maps in the derived category.^[26]

Puppe Sequence

The Puppe sequence provides a categorical framework for understanding fibrations at the level of spaces in homotopy theory, generalizing classical exact sequences to the homotopy category. For a Serre fibration p: E \to B with fiber F, the Puppe sequence is the infinite cofiber sequence F \to E \to B \to \Sigma F \to \Sigma E \to \Sigma B \to \Sigma^2 F \to \cdots, which is exact in the pointed homotopy category of topological spaces, meaning that at each term, the map is the homotopy cofiber of the previous map and the homotopy fiber of the next. This exactness ensures that the sequence captures the homotopical relationships between the fiber, total space, and base without truncation.^[27] The construction begins with the given fibration F \to E \xrightarrow{p} B, where F is taken as the homotopy fiber of p. The base B is then identified with the homotopy cofiber of the inclusion F \to E, up to homotopy equivalence, via the long exact sequence properties of fibrations. Iterating this process involves attaching the suspension \Sigma F as the homotopy cofiber of B \to * (or more precisely, connecting via the boundary map), and continuing indefinitely by suspending each term and forming successive homotopy cofibers. This augmentation yields the full exact sequence, preserving homotopical information across suspensions.^[28] In the stable homotopy category, where suspensions become invertible, the Puppe sequence specializes to a long exact triangle, reflecting the triangulated structure. Applying the Hurewicz homomorphism to this sequence relates it directly to homology, while taking homotopy groups \pi_* produces the long exact sequence of homotopy groups for the fibration; the brief reference here is that this long exact sequence arises as the \pi_*-version of the Puppe sequence itself. A key application of the Puppe sequence lies in the computation of Postnikov towers for simply connected spaces, where the tower is built as a sequence of principal fibrations with Eilenberg-MacLane spaces as fibers, and the k-invariants are precisely the connecting maps B \to \Sigma F from the Puppe sequence of each stage. This allows systematic approximation of a space by truncating its homotopy groups, facilitating calculations of homotopy classes and obstructions.^[29] In the broader context of model categories, Puppe exactness—requiring that the induced sequence in the homotopy category be exact at every position—characterizes fibrations in left proper model categories, providing a homotopical criterion equivalent to the right lifting property against acyclic cofibrations. This perspective unifies the treatment of fibrations across various homotopy-theoretic settings, such as simplicial sets or spectra.^[30]

Advanced Topics

Spectral Sequences

Spectral sequences provide powerful tools for computing the cohomology or homology of the total space of a fibration by relating it to the cohomology of the base and fiber through a filtered complex. In the context of a Serre fibration F \to E \xrightarrow{p} B, the Serre spectral sequence arises as a first-quadrant spectral sequence in cohomology, with E_2^{p,q} = H^p(B; \mathcal{H}^q(F)) converging to H^{p+q}(E), where \mathcal{H}^q(F) denotes the local coefficient system on B given by the action of \pi_1(B) on H^q(F). The differentials are d_r: E_r^{p,q} \to E_r^{p+r, q-r+1}. This sequence was introduced by Jean-Pierre Serre in his foundational work on the homology of fiber spaces. The construction of the Serre spectral sequence proceeds from the double complex associated to the skeletal filtration of the total space E, where the filtration is induced by the projection p: E \to B. Specifically, if B has a CW structure, one filters the cochain complex of E by the skeletons of B, leading to a double complex whose total cohomology yields the spectral sequence under suitable properness conditions, such as the fibration being orientable or the coefficients being a field. Convergence holds more generally by the Leray-Serre theorem, which guarantees that if both the base B and fiber F are simply connected, the spectral sequence converges strongly to the cohomology of E. The theorem also establishes edge homomorphisms from E_2^{p,0} \cong H^p(B) to H^p(E) and from E_2^{0,q} \cong H^q(F) to H^q(E), with the transgression maps relating higher differentials to the connecting homomorphisms in the long exact sequence of the fibration. A classic application is the computation of the cohomology of the Hopf fibration S^1 \to S^3 \to S^2 with integer coefficients. Since \pi_1(S^2) = 0, the local system \mathcal{H}^q(S^1; \mathbb{Z}) is trivial, but the spectral sequence reveals the structure through its E_2-page E_2^{p,q} = H^p(S^2; H^q(S^1; \mathbb{Z})), which is nonzero only for q = 0,1 and yields H^n(S^3; \mathbb{Z}) \cong \mathbb{Z} for n=0,3 after the differentials act, including a nontrivial d_2: E_2^{0,1} \to E_2^{2,0} that is an isomorphism; more generally, for projectivizations, twisted sheaves arise in analogous computations over non-simply connected bases. The Eilenberg-Moore spectral sequence extends these ideas to pullback diagrams, converging to the cohomology of the fiber of a pullback fibration, but its application is limited to situations where the fibration admits a suitable cotriple resolution, as developed in the original algebraic framework for limits in categories. This sequence often collapses to the long exact sequence when the higher pages degenerate immediately.^[31]

Orientability

A principal G-fibration p: E \to B, where G is a topological group acting freely and properly on the total space E, is orientable if it admits a reduction of the structure group to the subgroup G^+ consisting of orientation-preserving elements, equivalently if the associated principal G/G^+-bundle (the orientation bundle) admits a global section that is compatible with the G-action.^[32] This reduction ensures a consistent choice of orientation on the fibers across the base B.^[32] For circle bundles, which are principal S^1-fibrations, orientability corresponds to the structure group being the connected group U(1) of rotations rather than a larger group including reflections; the Euler class e(p) \in H^2(B; \mathbb{Z}) is defined for such oriented bundles (with structure group SO(2) ≅ U(1)) and measures the obstruction to the existence of a global section.^[33] In non-orientable cases, cohomology computations for the base or total space involve local coefficients twisted by the monodromy action of \pi_1(B) on the homology or cohomology of the fiber; specifically, the sheaf of local coefficients \mathcal{F} is the fiber's cohomology twisted by the representation \pi_1(B) \to \Aut(H^*(F; \mathbb{Z})), leading to H^*(B; \mathcal{F}). Orientable fibrations admit untwisted spectral sequences in their Serre or Eilenberg-Moore computations, where the E_2-term uses ordinary cohomology without local coefficients; this contrasts with non-orientable cases requiring twisted coefficients.^[33] For SO(n)-fibrations (already reduced from O(n)), further reduction to Spin(n) defines a spin structure, analogous to orientability but for double covers, enabling spin^c structures in the presence of additional line bundle data.^[32] The Hopf fibration S^1 \to S^3 \to S^2, as a principal U(1)-bundle, is orientable since its structure group is connected and its Euler class is the generator of H^2(S^2; \mathbb{Z}) \cong \mathbb{Z}, but the consistent orientation is preserved globally.^[32] In contrast, the circle bundle over \mathbb{RP}^2 obtained as the associated unit bundle to the tautological real line bundle (or the quotient of the Hopf fibration by antipodal action) is non-orientable, as the monodromy includes orientation-reversing elements, twisting the local coefficients in cohomology.^[33]

Euler Characteristic

In a Serre fibration F \to E \to B where F and B are finite CW-complexes with B connected, the Euler characteristic satisfies the multiplicative property \chi(E) = \chi(F) \chi(B).^[34] This holds more generally for fiber sequences of finitely dominated spaces, where the Euler characteristic is defined via the alternating trace on perfect chain complexes.^[35] A proof sketch proceeds via the Serre spectral sequence in homology with field coefficients, where the E_2-page is E_2^{p,q} = H_p(B; H_q(F; k)) and converges to H_{p+q}(E; k). The total rank on the E_2-page equals \chi(B) \chi(F), and since differentials increase filtration degree while preserving the total degree parity in a manner that preserves the alternating sum of ranks, the Euler characteristic of E equals the product.^[35] Alternatively, for CW-pairs, the result follows by induction on cells of B, as the preimage of each cell is homotopy equivalent to a product F \times D^n, whose chain complex tensors with that of the fiber, yielding multiplicativity via the Künneth theorem. This property is illustrated by the Hopf fibration S^1 \to S^3 \to S^2, where \chi(S^3) = 0 = \chi(S^1) \chi(S^2) = 0 \times 2, since \chi(S^n) = 1 + (-1)^n for the n-sphere. Similarly, the trivial S^1-bundle S^1 \times S^2 \to S^2 satisfies \chi(S^1 \times S^2) = 0 = \chi(S^1) \chi(S^2). In the oriented case, for a fibration with closed oriented fiber F over a closed oriented base B, the integrality of the Euler class in the cohomology of B ensures the formula holds without sign ambiguities, directly relating to \chi(F) via the Gysin sequence when F is a sphere. The multiplicative property generalizes to settings like complex K-theory and oriented cobordism, where Euler characteristics (or their refinements, such as Todd genus) multiply for oriented fibrations, reflecting the ring structures in these theories.^[36]

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