Fact-checked by Grok 2 weeks ago

Fiber bundle

In and , a fiber bundle is a fundamental consisting of a total space E, a base space B, and a surjective continuous p: E \to B known as the , such that for every point in B, there exists a neighborhood U where the preimage p^{-1}(U) is to the product U \times F via a fiber-preserving , with F denoting the —a that is locally constant over B. This local product ensures that fiber bundles generalize familiar objects like trivial bundles (where E \cong B \times F) and capture global twisting via transition functions defined on overlaps of the base's open cover, typically taking values in a G acting effectively on F. Fiber bundles encompass special cases such as vector bundles, where fibers are vector spaces and the structure group is a linear group like GL(n, \mathbb{R}), and principal bundles, where fibers are the structure group itself acting freely and transitively. They satisfy the homotopy lifting property with respect to all CW-complexes, making them Serre fibrations, and thus admit a long exact sequence in homotopy groups relating those of E, B, and F: \cdots \to \pi_n(F) \to \pi_n(E) \to \pi_n(B) \to \pi_{n-1}(F) \to \cdots. Classification of fiber bundles over a base B often proceeds via Čech cohomology H^1(B; G), where bundles are isomorphic if their transition cocycles are cohomologous, with universal bundles over classifying spaces BG providing a complete set of invariants. Developed in the mid-20th century through contributions from mathematicians like Norman Steenrod, , and others, fiber bundles arose from efforts to study fibrations and embeddings in , evolving from Whitney's work on immersions in the . Their importance lies in bridging local and global : they enable the construction of s, such as the Leray-Serre , to compute and of total spaces from base and fiber data, with applications to classes (e.g., Chern or Stiefel-Whitney classes for vector bundles), , and manifold invariants like the existence of vector fields or immersions. In , connections on fiber bundles define and curvature, underpinning gauge theories in physics, while in , they model loop spaces and higher structures like the S^1 \to S^3 \to S^2.

Introduction and History

Overview

In topology and geometry, a fiber bundle is a structure that generalizes the Cartesian product of spaces by allowing a "twisting" that varies continuously over a base space. It consists of a total space E, a base space B (often a manifold), a fiber F (a topological space attached to each point of B), and a continuous surjective projection map \pi: E \to B such that for every b \in B, the preimage \pi^{-1}(b) \cong F. This setup captures families of spaces parameterized by B, where each fiber is homeomorphic to F. Intuitively, fiber bundles represent "twisted products" of B and F: locally, over small open sets in B, the bundle resembles the direct product U \times F for U \subset B, but globally, the twisting may prevent a consistent with B \times F. This local-to-global distinction encodes topological obstructions, extending simpler constructions like covering spaces. Fiber bundles are foundational in , where they facilitate computations of groups and sequences relating the topology of E, B, and F; in , they describe the TM \to M over a manifold M, which models velocities and directions at each point of M. In physics, they underpin gauge theories, such as those for and Yang-Mills fields, where principal bundles with structure group like U(1) or SU(N) represent internal symmetries and connections define field strengths. A simple conceptual illustration is the trivial line bundle over the circle S^1, where B = S^1, F = \mathbb{R}, and E = S^1 \times \mathbb{R}; the projection \pi sends each point ( \theta, r ) \in E to \theta \in S^1, forming an infinite cylinder without twisting, unlike nontrivial variants.

Historical Development

The concept of fiber bundles emerged in the early 20th century amid advancements in topology and differential geometry, with foundational influences from Hassler Whitney's work in the 1930s on manifolds and embeddings. Whitney's 1935 paper introduced sphere spaces as generalizations of fiber spaces, providing key insights into local product structures and embedding theorems that foreshadowed bundle theory. These ideas built on earlier notions, such as Herbert Seifert's 1932-1933 fibrations of three-manifolds, but Whitney's contributions emphasized the topological decomposition of spaces into base and fiber components. The formal introduction of fiber bundles occurred in the 1940s through Jean Leray's pioneering efforts, developed during his internment in a prisoner-of-war camp from 1940 to 1945. In his captivity notes, Leray defined sheaves and fiber bundles to advance theory, addressing global topological properties via local data. This work, disseminated postwar, integrated bundles into , complemented by Charles Ehresmann and Jacques Feldbau's 1939-1941 generalizations incorporating structure groups and the covering property. By 1951, Norman Steenrod's The Topology of Fibre Bundles solidified these concepts as central to the field. Post-World War II developments in the 1950s advanced bundle theory through characteristic classes, notably by and . address and related publications established the differential geometry of fiber bundles, introducing intrinsic proofs for characteristic classes via and . on algebraic varieties extended these to homogeneous spaces, formalizing the Chern-Weil homomorphism for computing topological invariants from geometric data. In the 1960s, linked vector bundles to , with his 1961 collaboration with Friedrich Hirzebruch and subsequent 1963-1964 developments providing homotopy-theoretic classifications and applications to index theory. The theory evolved in the 1970s and 1980s toward generalizations, including and holomorphic bundles, driven by integrations with and . Ehresmann's earlier framework for structures matured into rigorous treatments of on bundles, while Atiyah's 1950s-1970s work on holomorphic bundles over curves extended to stability and moduli problems in . These advancements facilitated links to in physics, where bundles model fundamental interactions.

Core Definitions

Formal Definition

A fiber bundle is formally defined as a quadruple (E, B, F, \pi), where E and B are topological spaces, F is a fixed topological space known as the typical fiber, and \pi: E \to B is a continuous surjective map called the projection. For each point b \in B, the fiber over b is the preimage \pi^{-1}(b), which is homeomorphic to F. The total space E can be expressed as the disjoint union of all these fibers: E = \bigcup_{b \in B} \pi^{-1}(b). The defining property of a fiber bundle is local triviality. Specifically, for every b \in B, there exists an open neighborhood U \subseteq B of b and a \phi: \pi^{-1}(U) \to U \times F such that the following commutes: \begin{CD} \pi^{-1}(U) @>{\phi}>> U \times F \\ @V{\pi}VV @VV{\mathrm{pr}_1}V \\ U @= U \end{CD} where \mathrm{pr}_1: U \times F \to U is the onto the first , meaning \pi \circ \phi^{-1}(u, f) = u for all (u, f) \in U \times F. This condition ensures that the bundle is locally indistinguishable from a product bundle U \times F. When the base space B is paracompact, the local triviality condition allows for the existence of global sections under certain circumstances, as paracompactness guarantees partitions of unity that facilitate gluing local trivializations. The fiber over a single point b \in B is precisely \pi^{-1}(b) \cong F, highlighting how the bundle structure "fibers" the total space over the base. To see why local triviality enforces the fibered structure, note that the homeomorphisms \phi preserve the projection, so points in \pi^{-1}(U) are paired with base points in U exactly as in the product, allowing consistent identification of fibers across the base while permitting global twisting. This local product decomposition, combined with the surjectivity of \pi, ensures that E is structured as a family of copies of F parameterized by B.

Local Trivializations and Sections

A fiber bundle \pi: E \to B with fiber F is equipped with a local trivialization over an open cover \{U_\alpha\}_{\alpha \in I} of the base space B, consisting of homeomorphisms \phi_\alpha: \pi^{-1}(U_\alpha) \to U_\alpha \times F that are compatible with the projection, meaning \pi \circ \phi_\alpha^{-1}(x, f) = x for all (x, f) \in U_\alpha \times F. These trivializations ensure that the bundle structure is locally indistinguishable from a product bundle, allowing global properties to be analyzed through local product decompositions. The existence of such an open cover with local homeomorphisms to product spaces is a defining feature of fiber bundles, distinguishing them from more general fibrations. On overlaps U_\alpha \cap U_\beta, the maps are induced by the \phi_\beta \circ \phi_\alpha^{-1}: (U_\alpha \cap U_\beta) \times F \to (U_\alpha \cap U_\beta) \times F, which takes the form (x, f) \mapsto (x, g_{\alpha\beta}(x)(f)), where g_{\alpha\beta}: U_\alpha \cap U_\beta \to \mathrm{Homeo}(F) is a continuous map to the group of F. These maps satisfy the cocycle condition g_{\alpha\beta}(x) \circ g_{\beta\gamma}(x) = g_{\alpha\gamma}(x) on triple overlaps U_\alpha \cap U_\beta \cap U_\gamma, ensuring consistent gluing of the local trivializations to form the global bundle E. The maps encode the topological twisting of the bundle, determining its isomorphism class up to the choice of cover. A section of the bundle is a continuous map s: B \to E such that \pi \circ s = \mathrm{id}_B, selecting a unique point in each fiber \pi^{-1}(b) over every base point b \in B. In a local trivialization over U \subset B, a section restricts to s(x) = (x, \sigma(x)) for x \in U, where \sigma: U \to F is a continuous local section map. Trivial bundles always admit global sections, providing a continuous choice of points across all fibers. However, non-trivial bundles may also admit global sections; the converse—that a global section implies triviality—holds specifically for principal bundles. For bases B that are paracompact, the existence of sections can be established using subordinate to the trivializing cover \{U_\alpha\}. Specifically, if local sections over each U_\alpha can be extended or averaged compatibly via the \{\rho_\alpha\} with \sum \rho_\alpha = 1, a global section is constructed as s(b) = \sum_\alpha \rho_\alpha(b) \cdot s_\alpha(b), where s_\alpha is the local section over U_\alpha. This technique applies particularly to vector bundles over paracompact bases, guaranteeing at least one global section, and more generally to bundles where the admits suitable averaging or extension properties.

Structural Elements

Structure Groups and Transition Functions

In fiber bundles, the structure group G is a topological group that acts effectively on the fiber F, providing a framework for describing how local trivializations are related across the base space. This action ensures that the transition maps between overlapping trivializations take values in G, thereby encoding the twisting of the bundle in a group-theoretic manner. Given an open cover \{U_\alpha\}_{\alpha \in I} of the base space B and corresponding local trivializations \phi_\alpha: p^{-1}(U_\alpha) \to U_\alpha \times F, the transition functions g_{\alpha\beta}: U_\alpha \cap U_\beta \to [G](/page/G) are defined such that \phi_\beta \circ \phi_\alpha^{-1}(x, f) = (x, g_{\alpha\beta}(x) \cdot f) for x \in U_\alpha \cap U_\beta and f \in F, where \cdot denotes the right of [G](/page/G) on F. These functions capture the manner in which fibers are identified over overlaps, preserving the bundle's local product structure while allowing for global nontriviality. The transition functions satisfy the cocycle condition g_{\alpha\gamma}(x) = g_{\alpha\beta}(x) \cdot g_{\beta\gamma}(x) on triple overlaps U_\alpha \cap U_\beta \cap U_\gamma, ensuring consistency in the gluing process and forming a Čech 1-cocycle with respect to the sheaf of continuous [G](/page/Group)-valued functions on B. This condition guarantees that the resulting total space is well-defined as a . For a fixed open cover, the isomorphism classes of fiber bundles with fiber F and structure group G correspond bijectively to the cocycles in Z^1(\{U_\alpha\}, G), up to equivalence under coboundaries, which adjust the choice of local trivializations. More invariantly, the classification is given by the first Čech cohomology group \check{H}^1(B, G), where two cocycles are equivalent if they differ by a coboundary, reflecting changes in the trivializations. A bundle admits a reduction of its structure group to a G \subset \mathrm{Homeo}(F) if there exists an equivalent bundle where all functions take values in G; the isomorphism classes of such reduced bundles are then classified by the with coefficients in this smaller group. This reduction captures essential topological features of the bundle, such as or other symmetries preserved by the action.

Bundle Maps and Isomorphisms

A bundle map, or fiber-preserving map, between two fiber bundles \pi: E \to B with fiber F and \pi': E' \to B' with fiber F' consists of a pair of continuous maps \psi: E \to E' and f: B \to B' such that the diagram \begin{CD} E @>\psi>> E' \\ @V\pi VV @VV\pi'V \\ B @>f>> B' \end{CD} commutes, meaning \pi' \circ \psi = f \circ \pi. This condition ensures that \psi maps each fiber \pi^{-1}(b) over b \in B into the fiber \pi'^{-1}(f(b)) over f(b) \in B', preserving the fibration structure. In the context of fiber bundles with structure group G acting on the fibers, a bundle map \psi is called a fiber bundle if, in local trivializations, it restricts to fiber maps of the form F_b \to F'_{f(b)} that are equivariant with respect to the G-actions. For vector bundles, where fibers are vector spaces and G = GL(n, \mathbb{R}) or similar, these fiber maps are required to be linear. Such s form the arrows in the category of fiber bundles, enabling the study of relationships between different bundles over related bases. A bundle isomorphism is a bundle map (\psi, f) that is bijective on total spaces with a continuous inverse (\psi^{-1}, f^{-1}) which is also a bundle map, making f a of bases and \psi a fiberwise . Two fiber bundles over the same base are isomorphic if and only if their transition cocycles (with respect to compatible open covers) are cohomologous, meaning one cocycle differs from the other by a coboundary in the appropriate Čech cohomology group. Bundle isomorphisms preserve the structure group, as they conjugate transition functions within the group. Pullback bundles provide a canonical way to induce a new bundle from a between bases. Given a bundle \pi: E \to B with F and a continuous f: B' \to B, the f^*E \to B' has total space f^*E = \{(b', e) \in B' \times E \mid f(b') = \pi(e)\} and projection \pi_{f^*E}: f^*E \to B' defined by \pi_{f^*E}(b', e) = b'. The over b' \in B' is homeomorphic to the over f(b') in E, and the transition functions of f^*E are the compositions of those of E with f. There is a natural bundle \mathrm{pr}_2: f^*E \to E over f, given by \mathrm{pr}_2(b', e) = e, which identifies f^*E as the in the categorical sense. s classify bundles up to via to classifying spaces, but their primary role here is in constructing induced structures.

Examples and Constructions

Trivial and Product Bundles

A product bundle over a base space B with typical F is constructed directly as the total space E = B \times F, equipped with the map \pi: E \to B given by the first projection \pi(b, f) = b. This construction is always locally trivial, with each fiber \pi^{-1}(b) \cong F, and it serves as the standard model for triviality in fiber bundle theory. A fiber bundle (E, \pi, B) with fiber F is called trivial if it is bundle-isomorphic to the product bundle B \times F, meaning there exists a bundle map \phi: E \to B \times F that is a and satisfies \pi_B \circ \phi = \pi, where \pi_B is the projection onto B. Equivalently, a trivial bundle admits a global trivialization, a \phi: E \to B \times F compatible with the projections as above, which allows the bundle to be "untwisted" globally. Product bundles are inherently trivial by this definition, as the identity map provides the required isomorphism. The triviality of a fiber bundle can be characterized using its structure group G and transition functions. Specifically, given an open cover \{U_\alpha\} of B with local trivializations, the bundle is trivial if and only if there exists a choice of trivializations such that all transition functions g_{\alpha\beta}: U_\alpha \cap U_\beta \to G are constantly the identity element \mathrm{id} \in G. More generally, if G acts freely and simply transitively on F, the bundle is trivial precisely when the transition functions are globally constant, enabling a consistent global identification of fibers with F. This condition ensures the existence of a global trivialization, as the constant transitions allow patching local sections into a global one without twisting. A representative example of triviality arises when the base space B is contractible, such as \mathbb{R}^n or a point. In this case, any fiber bundle over B is trivial, as the contractibility implies a equivalence to a point, allowing the construction of a global section and subsequent trivialization via extension from the point fiber. Obstructions to the triviality of a fiber bundle are captured by characteristic classes in the cohomology of the base. For instance, in oriented vector bundles, the Euler class e(\xi) \in H^n(B; \mathbb{Z}) (where n is the bundle rank) vanishes if and only if the bundle admits a nowhere-zero global section, a necessary condition for triviality; a non-zero Euler class thus obstructs triviality. This class serves as a primary cohomological invariant measuring twisting in the bundle structure.

Nontrivial Topological Bundles

Nontrivial topological bundles are those that cannot be globally expressed as a product of the base and spaces, despite being locally trivial. Classic examples in low dimensions illustrate this twisting through transition functions that introduce a effect around the base. The and serve as fundamental instances of such bundles over the S^1, where the is either an or another , leading to non-orientable total spaces. The arises as a bundle with base S^1, the closed [-1,1], and total space the itself. A detailed proceeds via the quotient space [0, 2\pi] \times [-1,1] / \sim, where the identifies (0, y) \sim (2\pi, -y) for all y \in [-1,1], while points differing by multiples of $2\pi in the first coordinate are identified only through the base circle structure. The projection map \pi: [0, 2\pi] \times [-1,1] / \sim \to S^1 sends [\theta, y] to [\theta] in [0, 2\pi]/\{0 \sim 2\pi\}, ensuring each \pi^{-1}([\theta]) is homeomorphic to [-1,1]. To define this bundle via charts, cover the base S^1 with two open arcs U_0 = (-\pi/2, 3\pi/2)/ \sim and U_1 = (\pi/2, 5\pi/2)/ \sim, whose intersection consists of two connected components: one arc A and another B. Local trivializations over U_0 and U_1 are standard product charts, but the transition function g_{01}: U_0 \cap U_1 \to \mathrm{Homeo}([-1,1]) is given by g_{01}(x) = 1 () on A and g_{01}(x) = -1 (antipodal map, reflection through the origin) on B. This piecewise-defined twist encodes the global non-triviality, as the total space is non-orientable and lacks a consistent across the overlap. In contrast to the trivial cylinder bundle S^1 \times [-1,1], the Möbius strip's transition function prevents a global or consistent framing, manifesting the bundle's nontrivial class. Specifically, the clutching map induced by looping around S^1 lies in a non-trivial element of \pi_1(\mathrm{Homeo}([-1,1])) \cong \mathbb{Z}/2\mathbb{Z}, classifying it up to among bundles over S^1. This twist results in the total space being a non-orientable surface, where any attempt to traverse the base fully reverses the . The provides another exemplar, constructed as an S^1-bundle over S^1 with total space the , a compact non-orientable 2-manifold. Its structure mirrors the case but with fibers, incorporating a double twist: one from the base parametrization and another reflection in the fiber. Using a similar clutching construction, cover S^1 with two arcs as before; the function g_{01}: U_0 \cap U_1 \to \mathrm{Homeo}(S^1) acts as the identity on one overlap component and as multiplication by -1 (antipodal map on S^1 \subset \mathbb{C}, equivalent to reflection) on the other, often denoted g(\theta) = (-1, \phi \mapsto -\phi). This yields the quotient total space where fibers are glued with a combined orientation-reversing effect, distinguishing it from the trivial S^1 \times S^1. The resulting bundle is nontrivial because the functions generate a non-contractible loop in the structure group, leading to a total space without global sections preserving orientation and confirming its classification in \pi_1(\mathrm{Homeo}(S^1)) \cong \mathbb{Z}. Both examples highlight how nontriviality over S^1 stems from the fundamental group of the homeomorphism group of the fiber, where elements correspond to orientation-reversing twists that obstruct global trivializations. For these low-dimensional cases, the bundles are classified by the first \pi_1(\mathrm{Homeo}(F)), with the and Klein bottles representing the generators of the \mathbb{Z}/2\mathbb{Z} and \mathbb{Z} components, respectively, underscoring their roles as prototypes of twisted topological constructions.

Covering Spaces

A covering space is a fiber bundle \pi: E \to B where the fiber F over each point in the base space B is a discrete space, specifically a countable 0-dimensional manifold, and the projection \pi is a local homeomorphism, making the bundle locally trivial with discrete fibers. This structure ensures that each point in B has an evenly covered neighborhood, where the preimage under \pi consists of disjoint open sets each homeomorphic to that neighborhood via \pi. The universal of a path-connected, locally path-connected, and semilocally simply-connected B is a simply connected total space E such that the deck transformation group, consisting of homeomorphisms of E that commute with \pi, acts freely and properly on E and is isomorphic to the \pi_1(B). This group acts transitively on each fiber, reflecting the symmetries of the maximal unramified cover. In covering spaces, the transition functions are constant maps taking values in the automorphism group \Aut(F), which for a discrete fiber F is isomorphic to the group of permutations of the discrete set F. These functions ensure the local trivializations glue consistently over overlaps in an atlas for B, with the structure group being discrete and acting by permuting the sheets of the cover. The classification of connected covering spaces over a base B with basepoint corresponds bijectively to the conjugacy classes of s of \pi_1(B), where the subgroup associated to a is the image of \pi_1(E) under the induced map \pi_*. Specifically, n-sheeted connected s, those with fibers of n, correspond to subgroups of n in \pi_1(B). A representative example is the n-fold covering map S^1 \to S^1 given by z \mapsto z^n for z \in S^1 \subset \mathbb{C}, where the fiber over any point consists of n distinct n-th roots of unity, yielding a nontrivial bundle for n > 1 since the structure group acts nontrivially by cycling the sheets.

Vector and Principal Bundles

A vector bundle is a fiber bundle (E, p, B) where the typical fiber F is a vector space \mathbb{R}^n or \mathbb{C}^n, and the structure group is the general linear group GL(n, \mathbb{R}) or GL(n, \mathbb{C}), respectively, acting linearly on the fibers. The transition functions g_{\alpha\beta}: U_\alpha \cap U_\beta \to GL(n, \mathbb{R}) (or GL(n, \mathbb{C})) are linear isomorphisms that ensure the local trivializations glue compatibly while preserving the vector space structure on each fiber p^{-1}(b) \cong \mathbb{R}^n (or \mathbb{C}^n) for b \in B. This linear structure allows vector bundles to model tangent spaces and other linear approximations on manifolds. A principal G-bundle, for a topological or Lie group G, is a fiber bundle (P, p, B) with fiber G, equipped with a free and transitive right G-action that is fiber-preserving, meaning p(ug) = p(u) for u \in P and g \in G. The structure group is G itself, and transition functions take values in G, with local trivializations equivariant under this action. Associated to a principal G-bundle via a representation \rho: G \to GL(V) of G on a vector space V, one obtains a vector bundle with fiber V, constructed as the quotient (P \times V)/G where G acts diagonally. Key constructions include the frame bundle P(E) of a E with \mathbb{R}^n, which is the principal GL(n, \mathbb{R})-bundle whose sections correspond to bases (frames) in the fibers of E. Another example is the tautological over \mathbb{RP}^n, a real of with total \{(l, v) \in \mathbb{RP}^n \times \mathbb{R}^{n+1} \mid v \in l\}, where the over a line l is the line itself. Vector bundles are classified up to by clutching functions, which are the functions over the overlaps of a of the B, corresponding to in the group H^1(B, GL(n, \mathbb{R})) (or H^1(B, GL(n, \mathbb{C})) for complex bundles). For real vector bundles, the Stiefel-Whitney classes w_i(E) \in H^i(B; \mathbb{Z}/2\mathbb{Z}) provide complete invariants in many cases, capturing obstructions to (w_1) and other structural properties. A section of a vector bundle E \to B is a continuous map s: B \to E such that p \circ s = \mathrm{id}_B, which can be viewed as an \mathbb{R}^n-valued function on B using local trivializations. The zero section, defined by s_0(b) = 0_b where $0_b is the zero vector in the fiber over b, always exists and is unique.

Other Constructions

Sphere bundles arise as associated constructions to , where the fiber over each point in the base is the unit in the fiber. Given an n-dimensional real \xi: E \to B equipped with a fiberwise inner product, the bundle S(\xi): S(E) \to B is defined by taking the unit S^{n-1} in each E_b \cong \mathbb{R}^n, yielding a with structure group O(n). A canonical example is the unit ST S^n \to S^n, obtained as the bundle of the TS^n, where each is the unit S^{n-1} in the at points on the n-. Mapping tori provide another construction of fiber bundles over S^1. For a \phi: F \to F on a F, the mapping torus is the quotient space (S^1 \times F)/\sim, where ( \theta, f ) \sim ( \theta + 1, \phi(f) ) for \theta \in [0,1) and f \in F, projecting to S^1 with fiber F and monodromy given by \phi. This bundle is trivial if and only if \phi is isotopic to the ; otherwise, it exhibits nontrivial twisting, as seen in the Möbius band, which is the mapping torus of the antipodal map on [0,1]. Quotient bundles are formed by group actions preserving the fiber structure. If a group H acts freely on the total space E of a fiber bundle \pi: E \to B with fiber F, such that the action preserves fibers and commutes with the projection, the quotient map E/H \to B defines a fiber bundle with fiber F/H. For principal bundles, quotienting by a closed subgroup yields associated bundles with homogeneous fiber spaces. Prominent examples include the , viewed as a circle bundle S^1 \to S^3 \to S^2, which is the quotient of S^3 \subset \mathbb{C}^2 by the free U(1)-action (z,w) \mapsto (\lambda z, \lambda w) for \lambda \in U(1), projecting to the base S^2 \cong \mathbb{C}P^1 via [z:w]. Lens spaces L(n,q) arise as quotients S^{2n+1}/(\mathbb{Z}/q\mathbb{Z}), where \mathbb{Z}/q acts freely on S^{2n+1} \subset \mathbb{C}^{n+1} by by q-th roots of unity. This yields a circle bundle S^1 \to L(n,q) \to \mathbb{C}P^n, where L(n,q) is the total space and the fiber is S^1 from the residual S^1-action.

Differentiable and Smooth Bundles

Differentiable Fiber Bundles

A differentiable fiber bundle, also known as a fiber bundle, extends the topological notion to the category of smooth manifolds by imposing differentiability conditions on its structure maps. Specifically, it consists of a total space E, a base space B, and a typical fiber F, all of which are smooth manifolds, together with a smooth projection map \pi: E \to B that is a submersion, meaning the differential d\pi_e: T_e E \to T_{\pi(e)} B is surjective for every e \in E. This submersion property ensures that each fiber \pi^{-1}(b) for b \in B is a submanifold of E diffeomorphic to F, and it guarantees the existence of local smooth sections. The bundle is equipped with an atlas of local trivializations \{\phi_\alpha: \pi^{-1}(U_\alpha) \to U_\alpha \times F\}, where each U_\alpha is an open subset of B it, and each \phi_\alpha is a satisfying \pi \circ \phi_\alpha^{-1}(u, f) = u for u \in U_\alpha and f \in F. The functions g_{\alpha\beta}: U_\alpha \cap U_\beta \to \mathrm{Diff}(F), defined by \phi_\beta \circ \phi_\alpha^{-1}(u, f) = (u, g_{\alpha\beta}(u) \cdot f), are required to be maps. Here, \mathrm{Diff}(F) denotes the group of diffeomorphisms of F, which carries a manifold structure, often constructed via the that embeds F into to model the topology on diffeomorphisms. This compatibility of functions distinguishes differentiable bundles from their topological counterparts, which correspond to the C^0 case. More generally, one defines C^k-differentiable fiber bundles for k \in \mathbb{N} \cup \{\infty\}, where all manifolds, the projection, trivializations, and transition functions are C^k-smooth (with C^\infty corresponding to smooth bundles). For finite k, the Whitney embedding theorem ensures compatibility between C^k atlases by allowing smooth approximations, facilitating the extension to infinite differentiability when needed. At each point e \in E, the tangent space decomposes as T_e E = \ker(d\pi_e) \oplus H_e, where \ker(d\pi_e) is the vertical tangent space tangent to the fiber, and H_e is a complementary horizontal subspace (whose choice requires a connection in more advanced settings). A canonical example is the tangent bundle TM \to M of a smooth n-manifold M, where E = TM is the disjoint union of tangent spaces T_p M \cong \mathbb{R}^n, \pi projects a tangent vector to its base point, and local trivializations arise from coordinate charts on M with transition functions taking values in \mathrm{GL}(n, \mathbb{R}), the general linear group, which are smooth by the smoothness of coordinate changes. This structure makes TM a smooth vector bundle, a special case of differentiable fiber bundles with linear fibers.

Smooth Structures and Connections

In the context of smooth fiber bundles, where the total space E, base space B, and fiber F are manifolds and the transition functions are smooth maps, an Ehresmann provides a geometric structure that allows for the differentiation of sections and the definition of . Specifically, an Ehresmann on a smooth fiber bundle \pi: E \to B is defined by a smooth horizontal subbundle H \subset TE that is complementary to the vertical subbundle V_E = \ker(d\pi), satisfying TE = H \oplus V_E at each point, where d\pi: TE \to TB is the of the projection. This decomposition ensures that H projects isomorphically onto TB via d\pi, i.e., d\pi(H) = TB and H \cap V_E = \{0\}, enabling a consistent choice of "horizontal" directions transverse to the fibers. The primary utility of an Ehresmann connection lies in its ability to define parallel transport along curves in the base space. For a smooth curve \gamma: [0,1] \to B, parallel transport is given by the horizontal lift \tilde{\gamma}: [0,1] \to E such that \pi \circ \tilde{\gamma} = \gamma, \tilde{\gamma}(0) lies in a specified fiber \pi^{-1}(\gamma(0)), and the tangent vector \tilde{\gamma}'(t) remains horizontal (i.e., in H_{\tilde{\gamma}(t)}) for all t. This lift induces an isomorphism between the fibers over \gamma(0) and \gamma(1), preserving the bundle structure and allowing for the comparison of elements across different fibers. If the connection is integrable, meaning the horizontal distribution is Frobenius integrable, parallel transport becomes path-independent within horizontal leaves. Curvature quantifies the failure of the horizontal distribution to be integrable and measures the non-commutativity of around closed loops. For a general Ehresmann , the is captured by the vertical component of the bracket [H, H] \subset V_E, which vanishes the distribution is integrable. In the case of a principal G-bundle P \to B with structure group G, the is equivalently described by a algebra-valued 1-form \omega \in \Omega^1(P, \mathfrak{g}), and the is the 2-form \Omega = d\omega + \frac{1}{2}[\omega, \omega] \in \Omega^2(P, \mathfrak{g}), which takes values in the adjoint bundle \mathrm{ad}(P) = P \times_G \mathfrak{g}. The form \Omega determines the of the and plays a central role in gauging symmetries and computing characteristic classes. For vector bundles equipped with a Riemannian , metric connections generalize the from tangent bundles. A on a Riemannian vector (E, \langle \cdot, \cdot \rangle) over B is an Ehresmann that preserves the under , meaning that the inner product is constant along horizontal lifts of curves. The unique torsion-free , analogous to the , satisfies the Koszul formula adapted to bundle sections and ensures compatibility with the via \nabla \langle s, t \rangle = \langle \nabla s, t \rangle + \langle s, \nabla t \rangle for sections s, t. Such connections are crucial for defining geodesic sprays and minimizing lengths in the bundle context. In physics, connections on U(1)-bundles over parameter spaces provide the framework for the Berry connection, which arises in the adiabatic for . The Berry connection is a U(1)-valued 1-form A = i \langle n(\mathbf{R}) | \nabla_{\mathbf{R}} n(\mathbf{R}) \rangle on the parameter manifold \mathbf{R}, where |n(\mathbf{R})\rangle is an eigenstate of a H(\mathbf{R}), defining of phases along paths in parameter space. This connection's , the Berry curvature, governs topological phases and phenomena like the .

Generalizations and Applications

Topological and Algebraic Generalizations

introduced sheaf theory as a generalization of fiber bundles during his internment in Oflag XVIIA from 1940 to 1945, motivated by the need to handle local properties in fibrations. In this framework, for a map \pi: E \to B, the higher direct image sheaves \pi_* \mathcal{H}^p(E) assign to each U \subset B the cohomology groups H^p(\pi^{-1}(U)), forming a sheaf over the base B that captures how sections vary locally. This sheafification allows for the study of sections over the base as a sheaf, extending the global section space of ordinary fiber bundles to a more flexible structure that accounts for gluing conditions via the sheaf axiom. Fiber bundles can be generalized to allow singular fibers, where the fiber type varies over the base, relaxing the local triviality condition at certain points. In algebraic geometry, such generalizations appear in families of curves, where singular fibers may consist of nodal curves; for instance, in elliptic fibrations over a complex base, Kodaira classified singular fibers including type I_1 nodal rational curves, enabling the of degeneration in moduli spaces. In , fiber bundles generalize to stacky bundles, which are fibered categories over a satisfying descent data for effective gluing along covers. These stacks provide a framework for bundles with automorphisms, where the total space is itself a stack rather than a , allowing fibers to be categories rather than sets. A prominent example is gerbes, defined via as 2-stacks banded by a group like U(1); U(1)-gerbes generalize principal U(1)-bundles (line bundles) to a 2-categorical setting, classified by Dixmier-Douady classes in H^2(B, \mathbb{Z}). Principal bundles serve as the base case for these stacky constructions, with gerbes arising as associated 2-bundles. Holomorphic fiber bundles over complex manifolds, particularly vector bundles with structure group SL(n, \mathbb{C}), extend topological bundles by requiring holomorphic transition functions. These bundles are classified topologically by their Chern classes c_i \in H^{2i}(B, \mathbb{Z}) for i = 1, \dots, n, which are independent of the holomorphic structure and detect obstructions to triviality; for SL(n, \mathbb{C}), the first Chern class c_1 = 0 due to the special linear condition. The Chern classes arise from the curvature of a Hermitian metric via the Chern-Weil homomorphism, providing invariants in de Rham cohomology that match the topological ones. Infinite-dimensional fiber bundles arise in the study of loop groups and groups, where the structure group is infinite-dimensional, such as the loop group L U(n) consisting of smooth maps S^1 \to U(n). These bundles, termed loop bundles, classify families of finite-dimensional bundles parametrized by loop spaces and are analyzed using techniques from infinite-dimensional topology, including Grassmannians of Hilbert spaces. For groups \mathrm{Diff}(M) of a manifold M, associated bundles model reparametrizations in , with classification involving groups of infinite-dimensional groups.

Applications in Physics and Geometry

In , principal G-bundles over model the internal symmetries of fundamental interactions, where the on the bundle serves as the gauge potential A, a Lie algebra-valued 1-form that dictates of fields. The curvature form F = dA + A \wedge A of this connection represents the field strength tensor, and the Yang-Mills equations, \nabla \cdot F = J and D^* F = 0 in the vacuum, govern the dynamics of non-Abelian fields, unifying , weak, and strong forces in the . This framework, originally proposed in the context of isotopic spin invariance, has been rigorously formulated using fiber bundles to resolve issues of gauge invariance and quantization. In , the TM of provides the natural arena for velocities, with s at each point encoding 4-velocities along worldlines, while the pseudo-Riemannian g on M induces a on TM that facilitates the study of geodesics as horizontal lifts under the . Geodesics, defined by the equation \frac{D}{d\tau} \dot{\gamma} = 0 where \dot{\gamma} is the , represent the shortest paths in curved , crucial for describing free-falling observers and rays; the T^*M, dual to TM, similarly hosts momenta and forms the for formulations of . This bundle structure ensures covariance under diffeomorphisms, underpinning the . Spinor bundles arise as associated vector bundles to the double cover of the orthonormal frame bundle by the (n) group, enabling the formulation of the i \gamma^\mu \nabla_\mu \psi = m \psi on spin manifolds, where \nabla is the lifted from the and \psi are sections of the . The existence of such a lift, defining a , is obstructed by the second Stiefel-Whitney class w_2(TM) \in H^2(M; \mathbb{Z}/2\mathbb{Z}); if w_2(TM) = 0, the manifold admits , as required for supersymmetric theories and fermionic matter in . This topological condition determines whether Dirac spinors can be consistently defined globally, impacting models of and condensed matter. The S^3 \to S^2 with S^1 fibers exemplifies a U(1)- in , modeling fields where the connection's curvature corresponds to the monopole strength, leading to quantized flux \int_{S^2} F = 2\pi n. In the adiabatic approximation, transporting a spin-$1/2 particle around a closed loop on the base S^2 induces the \gamma = -m \Omega, with \Omega the subtended, arising as the of the U(1)- on the bundle of degenerate eigenspaces. This geometric phase manifests in phenomena like the and neutrino oscillations, linking bundle topology to observable interference effects. Characteristic classes of fiber bundles play a pivotal role in geometry, with the Euler class e(TM) of the tangent bundle evaluated on the fundamental class yielding the Euler characteristic \chi(M) = \langle e(TM), [M] \rangle, which by the Poincaré-Hopf theorem equals the sum of indices of zeros of any vector field on M. Thus, a nowhere-vanishing vector field exists if and only if \chi(M) = 0, providing a topological obstruction to parallelism on manifolds like the sphere S^2. In modern applications to topological insulators, K-theory classes of vector bundles over the Brillouin torus classify gapped phases, with the first Chern class c_1 quantifying the quantum Hall conductivity \sigma_H = \frac{e^2}{h} n via the integer n = \int_{T^2} c_1, as seen in recent models of fractional quantum Hall states and higher-order insulators.