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Vector bundle

In mathematics, particularly in algebraic topology and differential geometry, a vector bundle is a topological construction consisting of a total space E, a base space B, and a continuous surjection \pi: E \to B such that each fiber \pi^{-1}(b) over a point b \in B is a vector space (typically over \mathbb{R} or \mathbb{C}), and the bundle is locally trivial: the base B admits an open cover \{U_\alpha\} with homeomorphisms \phi_\alpha: \pi^{-1}(U_\alpha) \to U_\alpha \times \mathbb{R}^k (or \mathbb{C}^k) that restrict to linear isomorphisms on each fiber, for some fixed rank k.^[1] This local product structure ensures that vector addition and scalar multiplication are defined continuously across the total space, making vector bundles a natural way to "parametrize families of vector spaces" over a topological base.^[1] Vector bundles generalize tangent bundles on manifolds, where the fiber over each point is the tangent space, providing a framework to study how linear structures vary smoothly or continuously over a space.^[1] Key properties include the existence of a zero section (mapping each base point to the zero vector in its fiber), the ability to form direct sums and tensor products of bundles, and the notion of stable isomorphism, where two bundles E_1 and E_2 are stably isomorphic if E_1 \oplus \epsilon^n \cong E_2 \oplus \epsilon^n for some trivial bundle \epsilon^n and integer n.^[1] Isomorphisms between vector bundles preserve the projection and induce linear isomorphisms on fibers, while sections—continuous maps s: B \to E with \pi \circ s = \mathrm{id}_B—allow global selections of vectors from the fibers.^[1] The rank of a vector bundle, the dimension of its fibers, is locally constant, reflecting the bundle's uniformity.^[1] Classic examples illustrate the nontriviality of vector bundles: the trivial bundle B \times \mathbb{R}^k over any base B, which admits k nowhere-zero sections; the Möbius bundle, a non-orientable line bundle over the circle S^1 obtained as a quotient of [0,1] \times \mathbb{R}; and the tangent bundle TS^n of the n-sphere, which is trivial for n=1,3,7 but nontrivial otherwise, such as TS^2.^[1] Complex examples include the canonical line bundle over \mathbb{CP}^1 \cong S^2, whose dual is the hyperplane bundle, highlighting projective geometry connections.^[1] These constructions often arise via clutching functions on sphere products or pullbacks along maps to classifying spaces like the Grassmannians G_k(\mathbb{R}^\infty).^[1] Vector bundles play a central role in topology and geometry by encoding obstructions to triviality through characteristic classes, such as Stiefel-Whitney classes for real bundles (detecting orientability and immersion properties) and Chern classes for complex bundles (related to curvature in differential geometry).^[1] They underpin K-theory, where the Grothendieck group of isomorphism classes under direct sum yields invariants like the topological K-group K(X), linking to Bott periodicity and stable homotopy theory.^[1] Applications extend to principal bundles (via associated constructions), gauge theory in physics, and the study of manifolds' embeddability, making vector bundles indispensable for understanding global linear phenomena over nonlinear spaces.^[1]

Basic Definitions and Properties

Formal Definition

A vector bundle is formally defined as a triple (E, \pi, B), where B is a topological space serving as the base (typically assumed to be Hausdorff, second-countable, and paracompact to ensure desirable properties like the existence of partitions of unity), E is the total space (also a topological space), and \pi: E \to B is a continuous surjective map known as the bundle projection. For each point b \in B, the fiber \pi^{-1}(b) is a vector space over \mathbb{R} or \mathbb{C} that is isomorphic to a fixed vector space V of finite dimension n (called the rank of the bundle), and these isomorphisms preserve the vector space structure.^[1]^[2] The defining property of local triviality requires that there exists an open cover \{U_i\}_{i \in I} of B such that for each i, the restriction of the bundle over U_i is trivialized by a homeomorphism \phi_i: \pi^{-1}(U_i) \to U_i \times V satisfying \pi = \mathrm{pr}_1 \circ \phi_i (where \mathrm{pr}_1 is the projection onto the first factor), and such that the restriction of \phi_i to each fiber \pi^{-1}(b) for b \in U_i is a linear isomorphism \pi^{-1}(b) \to \{b\} \times V. These local trivializations ensure consistency across overlaps via transition functions taking values in the general linear group \mathrm{GL}(n, \mathbb{R}) or \mathrm{GL}(n, \mathbb{C}), though the details of such gluing are addressed separately. When B is a smooth manifold, the total space E and the trivializations are required to be smooth, making \pi a submersion and \phi_i diffeomorphisms.^[1]^[2] Vector space operations on the fibers—addition (v, w) \mapsto v + w (where v, w \in \pi^{-1}(b) for the same b) and scalar multiplication \lambda \cdot v for \lambda \in \mathbb{R} or \mathbb{C}—are defined fiberwise and extend to continuous (or smooth, in the manifold case) maps on the total space E, ensuring the bundle structure respects the linear algebra on each fiber.^[1]^[2] Classic examples illustrate this structure: the trivial bundle B \times V \to B with projection onto the first factor, where the total space is simply the product and global trivializations exist; the tangent bundle TM \to M of a smooth n-manifold M, where fibers are the tangent spaces T_b M \cong \mathbb{R}^n and local trivializations arise from charts on M; and the Möbius line bundle (rank 1) over the circle S^1, whose total space is homeomorphic to an open Möbius strip, demonstrating a non-trivial topology despite local triviality over suitable covers.^[1]^[2]

Transition Functions and Local Triviality

A vector bundle E \to B of rank n over a topological space B is locally trivial if there exists an open cover \{U_i\}_{i \in I} of B such that for each i, there is a homeomorphism \phi_i: \pi^{-1}(U_i) \to U_i \times \mathbb{R}^n satisfying \pi \circ \phi_i^{-1}(p, v) = p for all (p, v) \in U_i \times \mathbb{R}^n, and such that the restriction of \phi_i to each fiber \pi^{-1}(p) is a linear isomorphism onto \{p\} \times \mathbb{R}^n.^[3]^[4] These local trivializations allow the bundle to be pieced together from trivial bundles over the open sets U_i, with the linear structure preserved on fibers.^[5] On overlaps U_i \cap U_j, the transition functions g_{ij}: U_i \cap U_j \to \mathrm{GL}(n, \mathbb{R}) encode the compatibility between trivializations, defined such that \phi_j = (\mathrm{id}_{U_i \cap U_j}, g_{ij}) \circ \phi_i.^[3]^[4] These are smooth (or continuous, depending on the bundle's category) maps taking values in the general linear group, ensuring that the gluing respects the vector space structure.^[5] The collection \{g_{ij}\} must satisfy the cocycle condition: g_{ik} = g_{ij} g_{jk} on triple overlaps U_i \cap U_j \cap U_k, along with g_{ii} = \mathrm{id} and g_{ij} = g_{ji}^{-1}.^[4] This condition guarantees a well-defined global bundle without inconsistencies in the fiber identifications.^[3] Given such a cocycle \{g_{ij}\}, the total space E can be constructed explicitly as the quotient of the disjoint union \coprod_i U_i \times \mathbb{R}^n by the equivalence relation (p, v)_i \sim (p, g_{ij}(p) v)_j for p \in U_i \cap U_j.^[5] Conversely, any vector bundle admits such a presentation via transition functions on a suitable open cover.^[4] For paracompact bases B, the isomorphism classes of rank-n real vector bundles are in bijection with the first Čech cohomology group H^1(B, \mathrm{GL}(n, \mathbb{R})), where the cohomology class of the cocycle \{g_{ij}\} determines the bundle up to isomorphism.^[4] Two cocycles represent isomorphic bundles if they differ by a coboundary, corresponding to a change of trivializations. Local trivializations are unique up to homotopy in the sense that homotopies between cocycles yield isomorphic bundles, reflecting the topological invariance captured by the cohomology group.^[4]

Subbundles and Quotient Bundles

A subbundle of a vector bundle \pi: E \to X of rank n is a vector subbundle F \subset E of rank k \leq n, where F is itself a vector bundle over X via the restricted projection \pi|_F: F \to X, and the inclusion i: F \hookrightarrow E is a bundle morphism that is injective on each fiber F_x \hookrightarrow E_x.^[6]^[1] The rank k of the subbundle is the constant dimension of its fibers, which must be locally constant over X.^[6] Locally, over an open cover \{U_i\} of X, F|_{U_i} is a direct summand of E|_{U_i}, meaning E|_{U_i} \cong F|_{U_i} \oplus Q_i for some bundle Q_i over U_i, ensuring the subbundle inherits local triviality from E.^[6]^[2] For a subbundle F \subset E of constant rank, the quotient bundle E/F is the vector bundle of rank n - k with fibers E_x / F_x, and there is a short exact sequence of vector bundles

$0 \to F \to E \to E/F \to 0,

where the maps are the inclusion and the canonical projection, respectively.^[2]^[1] This sequence splits locally over the cover \{U_i\}, reflecting the direct sum decomposition; over paracompact bases, a global splitting exists.^[6] Trivial examples include the zero subbundle \{0\} \subset E, which has rank 0 and quotient isomorphic to E, and the full bundle E \subset E, with rank n and quotient the zero bundle.^[1] A nontrivial example arises in differential geometry: for a submanifold M \subset N of manifolds, the tangent bundle TM \subset TN|_M is a subbundle of rank \dim M, and the quotient (TN|_M)/TM is the normal bundle to M in N.^[2] More generally, integrable distributions on a manifold correspond to subbundles of the tangent bundle.^[2] The space of all rank-k subbundles of a fixed vector bundle E of rank n over X is parameterized by the Grassmannian bundle \mathrm{Gr}_k(E) \to X, whose fibers are Grassmannians \mathrm{Gr}_k(\mathbb{R}^n).^[1]

Morphisms and Equivalences

Vector Bundle Homomorphisms

A vector bundle homomorphism, also known as a morphism of vector bundles, between two vector bundles \xi = (E, \pi, B) and \xi' = (E', \pi', B') is a pair of continuous maps f: E \to E' and g: B \to B' such that \pi' \circ f = g \circ \pi, and for every b \in B, the restriction f_b: \pi^{-1}(b) \to \pi'^{-1}(g(b)) is a linear transformation between the corresponding fibers, which are vector spaces.^[7]^[8] This ensures that the map preserves the vector space structure fiberwise while commuting with the projections to the base spaces. In the smooth category, where the bundles are smooth over smooth manifolds, the maps f and g are required to be smooth.^[8] The base map g: B \to B' is uniquely determined by the total space map f, as g = \pi' \circ f \circ \pi^{-1} on the base, reflecting the induced action on the base space. Fiberwise linearity means that each f_b is a linear map, preserving addition and scalar multiplication within the fibers, which aligns with the local trivializations of the bundles where transition functions are linear isomorphisms.^[7]^[9] For a homomorphism f: E \to E' over the identity on the base (i.e., g = \mathrm{id}_B), the kernel \ker f at each point b \in B is defined as \{ v \in \pi^{-1}(b) \mid f(v) = 0 \}, forming a subspace of the fiber, and similarly the image \mathrm{im} f_b = f(\pi^{-1}(b)) is a subspace of \pi'^{-1}(b). If the rank of f—the dimension of the image on each fiber—is constant across the base, then by the constant rank theorem, \ker f and \mathrm{im} f assemble into vector subbundles of \xi and \xi', respectively.^[7]^[10] In this case, the homomorphism splits locally as a short exact sequence of vector bundles $0 \to \ker f \to \xi \to \mathrm{im} f \to 0. For surjective homomorphisms of constant rank equal to the rank of the target bundle, the map is locally an isomorphism onto its image, and similarly for injective maps.^[10] Examples of vector bundle homomorphisms include the inclusion map of a subbundle \eta \hookrightarrow \xi over the identity base map, where each fiber inclusion is linear by definition of a subbundle; the zero map, which sends every fiber to the zero subspace and has constant rank zero; and the identity map \mathrm{id}_\xi, which is linear on each fiber with constant full rank.^[8]^[9]

Isomorphisms and Stable Equivalence

An isomorphism between two vector bundles E \to B and F \to B over the same base space B is a bundle homomorphism \phi: E \to F that is bijective and admits an inverse bundle homomorphism \phi^{-1}: F \to E, such that both \phi and \phi^{-1} are linear isomorphisms on each fiber.^[1] Equivalently, \phi is a homeomorphism of total spaces that restricts to linear isomorphisms on fibers over corresponding base points.^[1] Two vector bundles over the same base are isomorphic if there exist local trivializations such that their transition functions g_{ij} and g'_{ij} are related by a coboundary, i.e., g'_{ij} = h_i g_{ij} h_j^{-1} for continuous maps h_i: U_i \to \mathrm{GL}(n, \mathbb{R}). Equivalently, this holds if their classifying maps to the Grassmannian \mathrm{Gr}(n, \mathbb{R}^\infty) are homotopic, preserving the vector space structure.^[1] A vector bundle E \to B of rank n is trivial, meaning isomorphic to the product bundle B \times \mathbb{R}^n \to B, if and only if it admits n global sections that are linearly independent on every fiber.^[1] The obstructions to triviality are captured by characteristic classes, such as the Stiefel-Whitney classes w_i(E) \in H^i(B; \mathbb{Z}/2\mathbb{Z}), where the bundle is orientable if w_1(E) = 0 (for line bundles over simply connected bases, this also implies triviality), and the bundle is trivial if all w_i(E) = 0 for i \geq 1.^[1] For example, the tangent bundle of the sphere S^n is trivial for n=1,3,7 but nontrivial otherwise. For even n, the nonzero Euler class provides an obstruction to triviality, while for odd n>7, other characteristic classes (such as certain Stiefel-Whitney classes) obstruct triviality.^[1] Stable equivalence provides a coarser notion of similarity between vector bundles, where two bundles E and F over B are stably equivalent if there exist integers k, m \geq 0 such that E \oplus \epsilon^k \cong F \oplus \epsilon^m, with \epsilon^1 = B \times \mathbb{R} denoting the trivial line bundle and \oplus the direct sum.^[1] The stable equivalence classes form the reduced K-group \tilde{K}(B), an abelian group under direct sum, where the operation is well-defined since adding trivial bundles does not change the isomorphism class in the stable range.^[1] Stiefel-Whitney classes are invariant under stable equivalence, providing topological invariants for these classes.^[1] The clutching construction illustrates stable equivalence particularly for bundles over spheres: a rank-n vector bundle over S^r is determined up to isomorphism by a clutching function f: S^{r-1} \to \mathrm{GL}(n, \mathbb{R}), obtained by gluing trivial bundles over the upper and lower hemispheres via f, and two such bundles are stably equivalent if their clutching functions become homotopic after stabilizing with trivial bundles of sufficiently high rank.^[1] For instance, over S^2, clutching functions in [\pi_1(\mathrm{U}(n))] classify complex bundles, and stable equivalence corresponds to maps into the stable unitary group, linking to Bott periodicity in K-theory.^[1]

Sections and Sheaf Perspectives

Global and Local Sections

A section of a vector bundle \pi: E \to B is a continuous map s: B \to E such that \pi \circ s = \mathrm{id}_B, meaning that for every point b \in B, s(b) lies in the fiber \pi^{-1}(b).^[11] Such a map assigns to each base point an element of the corresponding fiber in a continuous manner. A global section is one defined over the entire base space B, while a local section is defined over an open subset U \subseteq B, with its image contained in \pi^{-1}(U).^[11] The set of sections over a fixed open set inherits a vector space structure from the fibers: for two sections s_1, s_2 over U and a scalar \lambda, the pointwise sum (s_1 + s_2)(b) = s_1(b) + s_2(b) and scalar multiple (\lambda s_1)(b) = \lambda s_1(b) are well-defined since all fibers are isomorphic to the same vector space.^[11] The space of all global sections, denoted \Gamma(E), forms a module over the ring C^0(B) of continuous real-valued functions on B, with the module action given by (\phi \cdot s)(b) = \phi(b) s(b) for \phi \in C^0(B) and s \in \Gamma(E).^[11] Every vector bundle admits a zero section, the global section z: B \to E defined by z(b) = 0 in each fiber \pi^{-1}(b).^[11] In a trivial bundle E \cong B \times V, constant sections exist, corresponding to maps s_v(b) = v for a fixed vector v \in V.^[11] Non-trivial bundles may lack non-zero global sections, illustrating topological obstructions. For example, the tautological line bundle over \mathbb{CP}^1 \cong S^2, which is the associated vector bundle to the Hopf fibration S^3 \to S^2, admits no global non-vanishing sections.^[12] This follows from the fact that a line bundle is trivial if and only if it has a global non-zero section.^[12] Thus, it admits only the zero global section, and the space of global sections is the zero vector space (dimension 0).

Locally Free Sheaves

In algebraic geometry, there is a natural equivalence between vector bundles over a scheme B and certain sheaves of \mathcal{O}_B-modules on B. Specifically, given a vector bundle E \to B of rank n, one associates the sheaf \tilde{E} of \mathcal{O}_B-modules defined by \tilde{E}(U) = \Gamma(\pi^{-1}(U), E) for each open subset U \subseteq B, where \pi: E \to B is the projection morphism and \Gamma denotes the module of sections of E over the preimage \pi^{-1}(U). This construction yields a coherent sheaf \tilde{E} that is locally free of constant rank n, and the assignment E \mapsto \tilde{E} establishes an equivalence of categories between vector bundles over B and finite locally free sheaves of \mathcal{O}_B-modules.^[13] A sheaf \mathcal{F} of \mathcal{O}_X-modules on a scheme X is called locally free of rank n if, for every point x \in X, there exists an open neighborhood U \ni x such that the restriction \mathcal{F}|_U is isomorphic to the direct sum of n copies of the structure sheaf \mathcal{O}_U, i.e., \mathcal{F}|_U \cong \mathcal{O}_U^{\oplus n}. Equivalently, each stalk \mathcal{F}_x is a free \mathcal{O}_{X,x}-module of rank n, generated by n elements that form a local frame, mirroring the local trivializations of the corresponding vector bundle. The rank function \rho(\mathcal{F}): X \to \mathbb{Z}_{\geq 0}, defined by \rho(\mathcal{F})(x) = \rank_{\mathcal{O}_{X,x}}(\mathcal{F}_x), is locally constant on X, and for sheaves arising from vector bundles, it takes a constant value equal to the bundle's rank.^[14] Representative examples illustrate this correspondence. The trivial line bundle over B, which is isomorphic to B \times k \to B for a field k, corresponds to the structure sheaf \mathcal{O}_B, a locally free sheaf of rank 1 whose stalks are free of rank 1 everywhere. On a smooth variety X over a field, the tangent sheaf T_X = \Hom_{\mathcal{O}_X}(\Omega_{X/k}, \mathcal{O}_X), dual to the sheaf of Kähler differentials, is locally free of rank equal to \dim X, reflecting the local triviality of the tangent bundle.^[14] This sheaf perspective preserves functoriality with respect to base change. For a morphism of schemes g: B' \to B, the pullback sheaf g^* \tilde{E} is defined by (g^* \tilde{E})(V) = \tilde{E}(g(V)) for open V \subseteq B', and it corresponds to the pullback vector bundle E' = E \times_B B' \to B', ensuring that the equivalence respects pullbacks. Global sections of the sheaf recover those of the bundle, with \Gamma(B, \tilde{E}) = H^0(B, \tilde{E}) consisting of sections over the total space.^[13]

Algebraic Operations

Direct Sums and Tensor Products

The direct sum of two vector bundles E \to B and F \to B over the same base space B is the vector bundle E \oplus F \to B defined by the total space \{(e, f) \in E \times F \mid \pi_E(e) = \pi_F(f)\}, where \pi_E and \pi_F are the respective projections to B, and the projection \pi: E \oplus F \to B given by \pi(e, f) = \pi_E(e). The fiber over each b \in B is the direct sum of the fibers E_b \oplus F_b. This construction extends associatively and commutatively to finite direct sums of bundles, with the zero bundle serving as the identity element.^[1]^[15] To ensure compatibility with the bundle structure, the direct sum is constructed via local trivializations: if E|_U \cong U \times V and F|_U \cong U \times W over an open cover \{U_i\} of B, then (E \oplus F)|_U \cong U \times (V \oplus W), with transition functions forming block-diagonal matrices using those of E and F. The rank of E \oplus F is the sum of the ranks, \mathrm{rank}(E \oplus F) = \mathrm{rank}(E) + \mathrm{rank}(F). A representative example is the Whitney sum of the tangent bundle TM and normal bundle NM to the n-sphere S^n \subset \mathbb{R}^{n+1}, which yields the trivial bundle S^n \times \mathbb{R}^{n+1}.^[16]^[2]^[1] The tensor product E \otimes F \to B of vector bundles E \to B and F \to B has total space \bigsqcup_{b \in B} (E_b \otimes F_b), equipped with the finest topology making the projections and inclusions continuous, and projection \pi: E \otimes F \to B defined fiberwise. The fiber over b \in B is E_b \otimes F_b. Locally, if E|_U \cong U \times V and F|_U \cong U \times W, then (E \otimes F)|_U \cong U \times (V \otimes W), with transition functions given by the tensor products g^E_{ij} \otimes g^F_{ij} of those for E and F. The rank is the product \mathrm{rank}(E \otimes F) = \mathrm{rank}(E) \cdot \mathrm{rank}(F). This operation is associative, commutative, and distributive over direct sums. Examples include the exterior powers \Lambda^k E, obtained as quotients of iterated tensor products by the alternating relations, which form bundles of rank \binom{\mathrm{rank}(E)}{k}.^[16]^[2]^[1] The direct sum satisfies the universal property that vector bundle homomorphisms from E \oplus F to another bundle G correspond bijectively to pairs of homomorphisms from E to G and F to G. The tensor product satisfies the universal property for bilinear maps: for any vector bundle G \to B, the bundle homomorphisms E \otimes F \to G correspond to bilinear maps of bundles E \times F \to G that are linear in each factor over B. These properties mirror those in the category of vector spaces and ensure the operations are well-behaved in the category of vector bundles.^[16]^[1]

Dual Bundles

The dual bundle of a vector bundle E \to B over a base space B (with real or complex fibers) is the vector bundle E^* \to B whose fiber over each point x \in B is the dual vector space (E_x)^* = \Hom(V_x, \mathbb{R}) (or \Hom(V_x, \mathbb{C})), consisting of all linear functionals on the fiber E_x.^[17] The total space E^* is equipped with the finest topology making the projection \pi_{E^*}: E^* \to B continuous and ensuring local trivializations are homeomorphisms (or diffeomorphisms in the smooth case).^[16] If \{U_i\} is an open cover of B with local trivializations \psi_i: \pi_E^{-1}(U_i) \to U_i \times \mathbb{R}^k for E of rank k, then the transition functions for E^* are given by g_{ij}^{E^*}(x) = (g_{ij}^E(x))^{-T}, the negative transpose of those for E, ensuring compatibility on overlaps U_{ij} = U_i \cap U_j.^[17] For vector bundles E \to B and F \to B, the Hom bundle \Hom(E, F) \to B has fibers \Hom(E_x, F_x) over each x \in B, comprising linear maps between fibers, and its sections over open sets correspond to vector bundle homomorphisms from E to F that are fiberwise linear.^[16] This bundle is isomorphic to E^* \otimes F, where the tensor product is the algebraic operation on bundles (built fiberwise and glued via transition functions g_{ij}^{E^* \otimes F} = g_{ij}^{E^*} \otimes g_{ij}^F).^[18] A canonical evaluation map \ev: E \times_B E^* \to \underline{\mathbb{R}}_B exists, where \underline{\mathbb{R}}_B denotes the trivial line bundle over B with fiber \mathbb{R} (or \mathbb{C}), defined fiberwise by \ev_x(v_x, \phi_x) = \phi_x(v_x) for v_x \in E_x and \phi_x \in (E_x)^*.^[19] This bilinear pairing induces a nondegenerate duality between E and E^*. Prominent examples include the cotangent bundle T^*M of a smooth manifold M, which is the dual bundle (TM)^* to the tangent bundle TM \to M, with sections being differential 1-forms.^[17] Another is the determinant line bundle \det E = \bigwedge^k E for a rank-k bundle E, the top exterior power, which is a line bundle whose dual is \det E^* = \bigwedge^k E^* \cong (\bigwedge^k E)^*.^[16] The duality yields a canonical pairing on global sections \langle \cdot, \cdot \rangle: \Gamma(E) \times \Gamma(E^*) \to C^0(B), mapping continuous sections s \in \Gamma(E) and \sigma \in \Gamma(E^*) to the continuous function x \mapsto \sigma_x(s_x) on the base B.^[19] This pairing is bilinear and extends the fiberwise evaluation, facilitating integrations and other operations in geometry and analysis.^[16]

Geometric and Topological Structures

Pullbacks and Pushforwards

In the context of vector bundles, the pullback operation allows one to induce a new vector bundle over a different base space via a continuous map between bases. Given a vector bundle \pi: E \to B over a topological space B and a continuous map f: B' \to B, the pullback bundle f^* E \to B' is defined with total space f^* E = \{ (b', e) \in B' \times E \mid f(b') = \pi(e) \}, equipped with the projection \pi_{f^* E}: f^* E \to B' given by (b', e) \mapsto b'.^[20]^[21] The fiber over a point b' \in B' is (f^* E)_{b'} = E_{f(b')}, which is naturally isomorphic to the fiber E_{f(b')} of the original bundle as vector spaces.^[20]^[22] To verify that f^* E forms a vector bundle, local trivializations are constructed from those of E. If \phi_i: \pi^{-1}(U_i) \to U_i \times \mathbb{R}^n are local trivializations over an open cover \{U_i\} of B with transition functions g_{ij}: U_i \cap U_j \to \mathrm{GL}_n(\mathbb{R}), then for open sets V_k = f^{-1}(U_k) covering B', the pullback trivializations are \phi'_k: (\pi_{f^* E})^{-1}(V_k) \to V_k \times \mathbb{R}^n defined by \phi'_k(b', e) = (b', \phi_k(e)), and the induced transition functions are (f^* g_{ij})(b') = g_{ij}(f(b')) for b' \in V_i \cap V_j.^[20]^[21] These transition functions ensure the compatibility required for a vector bundle structure, and the construction is unique up to isomorphism.^[21] The pullback functor f^* is natural and preserves algebraic operations on vector bundles. Specifically, for vector bundles E_1, E_2 over B, it satisfies f^*(E_1 \oplus E_2) \cong f^* E_1 \oplus f^* E_2 and f^*(E_1 \otimes E_2) \cong f^* E_1 \otimes f^* E_2, with the isomorphisms induced pointwise on fibers.^[21] Moreover, for composable maps g: B'' \to B' and f: B' \to B, the functoriality holds: (f \circ g)^* E \cong g^* (f^* E).^[21] The pushforward, or direct image, f_* E of a vector bundle \pi: E \to B' along a continuous map f: B' \to B is primarily defined at the level of sheaves of sections rather than as a bundle itself. For an open set V \subset B, the sections over V are \Gamma(f_* E, V) = \Gamma(E, f^{-1}(V)), where sections of E over f^{-1}(V) are those of the restriction E|_{f^{-1}(V)} \to f^{-1}(V).^[23] This sheaf f_* E is a sheaf of \mathcal{O}_B-modules if E corresponds to a locally free sheaf, but it is locally free (hence represents a vector bundle) only under additional conditions on f, such as when f is proper with finite fibers.^[24] In such cases, particularly for finite covering maps, the fiber over b \in B can be taken as the direct sum \bigoplus_{b' \in f^{-1}(b)} E_{b'}, yielding a vector bundle structure.^[25] Examples of pullbacks include the induced bundle on a submanifold: if i: M' \hookrightarrow M is the inclusion of a submanifold, then i^* (TM) = TM', the tangent bundle restricted to M'.^[21] Another instance is the restriction to fibers: for a bundle \pi: E \to B, the pullback \pi^* E over E has fibers over e \in E isomorphic to the fiber E_{\pi(e)}.^[20] For pushforwards, consider a finite covering p: \tilde{B} \to B; then p_* \mathcal{O}_{\tilde{B}} is locally free of rank equal to the degree of the cover, corresponding to a vector bundle whose sections over an open set V \subset B are the continuous functions on p^{-1}(V) \subset \tilde{B}.^[23]

Principal Bundles and Associated Vector Bundles

A principal G-bundle over a base space B is a fiber bundle \pi: P \to B equipped with a continuous free right action of a topological group G on P satisfying \pi(p g) = \pi(p) for all p \in P and g \in G, such that the orbit map P \to P/G identifies B with the quotient space P/G.^[26] This action ensures that each fiber \pi^{-1}(b) is equivariantly homeomorphic to G, with G acting on itself by right multiplication.^[27] Locally, principal G-bundles are trivial: there exists an open cover \{U_\alpha\} of B with equivariant homeomorphisms \phi_\alpha: \pi^{-1}(U_\alpha) \to U_\alpha \times G satisfying \phi_\alpha(p g) = (\pi(p), g') where g' \in G aligns with the action, and these trivializations are compatible on overlaps via transition functions g_{\alpha\beta}: U_\alpha \cap U_\beta \to G.^[26] Given a principal G-bundle \pi: P \to B and a continuous representation \rho: G \to \mathrm{GL}(V) of G on a vector space V, the associated vector bundle is constructed as the quotient space E = (P \times V)/G \to B, where G acts diagonally via (p, v) \cdot g = (p g, \rho(g^{-1}) v) for p \in P, v \in V, and g \in G.^[1] The projection map sends the equivalence class [p, v] to \pi(p) \in B, yielding fibers homeomorphic to V with a natural vector space structure induced by the representation.^[28] The transition functions of E over overlaps U_\alpha \cap U_\beta are then given by \rho(g_{\alpha\beta}^P), where g_{\alpha\beta}^P are the transition functions of the principal bundle P, ensuring that local trivializations of E over U_\alpha are U_\alpha \times V glued compatibly via these linear maps.^[1] A canonical example arises from any rank-n vector bundle \xi: E \to B, whose frame bundle \mathrm{Fr}(\xi) is the principal \mathrm{GL}(n, \mathbb{R})-bundle over B with fiber over b \in B consisting of all ordered bases (frames) of the fiber E_b \cong \mathbb{R}^n.^[27] The right \mathrm{GL}(n, \mathbb{R})-action changes bases via matrix multiplication, and local trivializations follow from those of \xi by mapping frames to standard bases in \mathbb{R}^n.^[1] If \xi is equipped with a Riemannian metric, the orthonormal frame bundle reduces to a principal O(n)-bundle, where fibers consist of orthonormal bases, obtained by restricting the structure group from \mathrm{GL}(n, \mathbb{R}) to its subgroup O(n) via the standard orthogonal representation.^[27] Conversely, every rank-n vector bundle \xi: E \to B determines a unique principal \mathrm{GL}(n, \mathbb{R})-bundle, namely its frame bundle \mathrm{Fr}(\xi), and \xi recovers as the associated vector bundle \mathrm{Fr}(\xi) \times_{\mathrm{GL}(n, \mathbb{R})} \mathbb{R}^n under the standard representation of \mathrm{GL}(n, \mathbb{R}) on \mathbb{R}^n.^[27] This equivalence shows that vector bundles and principal \mathrm{GL}(n, \mathbb{R})-bundles over the same base are in bijective correspondence, with the association functor preserving bundle isomorphisms.^[1]

Differentiable Vector Bundles

Smooth Vector Bundles

A smooth vector bundle over a smooth manifold M refines the topological notion by requiring that the transition functions g_{ij}: U_i \cap U_j \to \mathrm{GL}(n, \mathbb{R}), where \{U_i\} is an open cover of M and the g_{ij} satisfy the cocycle condition g_{ij} g_{jk} = g_{ik}, are smooth maps compatible with the smooth atlas of M. This compatibility ensures that the total space E inherits a smooth manifold structure of dimension \dim M + n, the projection \pi: E \to M is a smooth submersion, and local trivializations \phi_i: \pi^{-1}(U_i) \to U_i \times \mathbb{R}^n are smooth diffeomorphisms preserving the vector space structure on each fiber. Such bundles capture families of vector spaces that vary smoothly along M, enabling differential geometric constructions on the total space.^[4] The space of smooth sections \Gamma^\infty(E) comprises all smooth maps s: M \to E satisfying \pi \circ s = \mathrm{id}_M, which locally correspond to smooth functions M \to \mathbb{R}^n via the trivializations. This space forms a module over C^\infty(M) and carries a natural Fréchet topology induced by seminorms measuring suprema of derivatives of local representatives, rendering \Gamma^\infty(E) a Fréchet manifold of infinite dimension. Smooth sections thus provide a framework for defining differential operators and integrals on the bundle.^[4] Representative examples illustrate these features. The tangent bundle TM of a smooth manifold M is a smooth vector bundle of rank \dim M, with transition functions given by the Jacobians of coordinate change maps, which are smooth by definition of the manifold structure. Smooth line bundles over surfaces, such as the hyperplane bundle over \mathbb{CP}^1 (or S^2), arise from clutching constructions with smooth transition functions like g(z) = z/|z| on the equator, yielding non-trivial topology while preserving smoothness.^[4]^[8] Partitions of unity subordinate to the cover \{U_i\} on paracompact manifolds like M enable global integration of compactly supported sections over the fibers: locally, a section s integrates fiberwise via the Lebesgue measure on \mathbb{R}^n, and these are glued smoothly using the partition to define \int_E |s|^2 \, d\mathrm{vol} or similar fiber densities. For smooth classification, the Whitney embedding theorem embeds M into Euclidean space, allowing reduction of bundle structures to those over \mathbb{R}^k via smooth extensions, which classifies isomorphism classes up to stable equivalence through clutching functions on spheres.^[4]^[8]

Connections on Vector Bundles

In the context of smooth vector bundles over a smooth manifold M, a connection provides a means to differentiate sections along vector fields, enabling the definition of parallel transport and covariant derivatives that respect the bundle's geometry.^[29] Formally, a connection \nabla on a smooth vector bundle E \to M is a bilinear map \nabla: \Gamma(TM) \times \Gamma(E) \to \Gamma(E), written (X, s) \mapsto \nabla_X s, where X is a smooth vector field on M and s is a smooth section of E.^[29] It satisfies the Leibniz rule: for any smooth function f \in C^\infty(M), \nabla_X (f s) = (X f) s + f \nabla_X s.^[29] This rule ensures that the connection acts linearly over the structure sheaf while accounting for the directional derivative of scalar coefficients.^[29] Locally, over a coordinate chart U \subset M with a local frame \{e_1, \dots, e_r\} for E|_U, the connection takes the form \nabla = d + A, where d is the exterior derivative on forms with values in E|_U, and A is a connection 1-form taking values in \Omega^1(U, \End(E)), the space of endomorphism-valued 1-forms.^[30] For a section s = \sum_i f^i e_i with f^i \in C^\infty(U), the local covariant derivative is \nabla s = \sum_i e_i \otimes df^i + \sum_{i,j} f^i A^j_i \otimes e_j, where A = (A^j_i) is the matrix of 1-forms.^[30] Under a change of frame via a GL(r, \mathbb{R})-valued transition function g: U \to GL(r, \mathbb{R}), the connection form transforms as \tilde{A} = g^{-1} A g + g^{-1} dg, preserving the global structure.^[30] Parallel transport induced by \nabla allows the comparison of fibers along smooth curves \gamma: [0,1] \to M. For a vector v \in E_{\gamma(0)}, the parallel transport to E_{\gamma(1)} is the unique solution to the ordinary differential equation (ODE) \frac{d}{dt} s(t) + A_{\gamma(t)}(s(t)) \cdot \dot{\gamma}(t) = 0, where s(t) \in E_{\gamma(t)} with s(0) = v.^[29] This defines a linear isomorphism P^\nabla_\gamma: E_{\gamma(0)} \to E_{\gamma(1)}, which is independent of parametrization and satisfies the group law for concatenated curves, forming the holonomy representation.^[29] The curvature of a connection \nabla measures its deviation from flatness and is given by the 2-form \Omega^\nabla = dA + A \wedge A \in \Omega^2(M, \End(E)), where the wedge product incorporates the Lie bracket in \End(E).^[29] Locally, for vector fields X, Y, \Omega^\nabla(X, Y) s = \nabla_X \nabla_Y s - \nabla_Y \nabla_X s - \nabla_{[X,Y]} s, acting as an endomorphism on sections.^[29] The second Bianchi identity holds: d\Omega + [A, \Omega] = 0, where [ \cdot, \cdot ] denotes the graded commutator, ensuring the curvature satisfies a consistency condition under parallel transport.^[29] Examples illustrate these concepts concretely. On the tangent bundle TM \to M of a Riemannian manifold (M, g), the Levi-Civita connection is the unique torsion-free connection satisfying \nabla_X Y \cdot Z = X(Y \cdot Z) for all vector fields X, Y, Z, preserving the metric and enabling geodesic flow.^[29] For a trivial bundle E = M \times V \to M with global frame \{e_i\}, the trivial connection is \nabla_X s = X(s), where s is viewed as a V-valued function, yielding zero curvature \Omega = 0.^[29]

Applications in Topology and Geometry

Characteristic Classes

Characteristic classes are topological invariants associated to vector bundles that capture obstructions to the existence of sections or provide cohomology classes measuring the bundle's twisting relative to the trivial bundle. For a vector bundle E \to B over a paracompact base space B, these classes live in the cohomology ring of B and are natural under bundle maps, making them useful for classifying bundles up to isomorphism. They were developed in the 1930s and 1940s through axiomatic approaches that emphasize functoriality and additivity under direct sums. For complex vector bundles, the Chern classes c_k(E) \in H^{2k}(B; \mathbb{Z}) form a sequence of cohomology classes, with the total Chern class defined as c(E) = 1 + c_1(E) + c_2(E) + \cdots + c_r(E), where r is the rank of E, and higher classes vanish. These classes satisfy the axioms of naturality, meaning that for a continuous map f: X \to B, f^* c_k(E) = c_k(f^* E), and the Whitney sum formula c(E \oplus F) = c(E) \cup c(F) for bundles E, F \to B. The Chern classes originate from Shiing-Shen Chern's work on Hermitian manifolds and were axiomatized to apply topologically to all complex bundles. For real vector bundles, the Stiefel-Whitney classes w_k(E) \in H^k(B; \mathbb{Z}/2) provide analogous invariants, with the total class w(E) = 1 + w_1(E) + \cdots + w_r(E). They satisfy the same naturality axiom and Whitney sum formula w(E \oplus F) = w(E) \cup w(F), and were introduced independently by Eduard Stiefel and Hassler Whitney as obstructions to extending frames over skeleta. On manifolds, the Stiefel-Whitney classes of the tangent bundle obey Wu's formula, relating them to the action of Steenrod squares on the Wu class \nu, via w(TM) = \mathrm{Sq}(\nu), which determines the orientability and other properties. The Euler class e(E) \in H^n(B; \mathbb{Z}) is defined for an oriented real vector bundle E of rank n, serving as the top characteristic class and vanishing if E admits a nowhere-zero section. For oriented rank-2 bundles, it coincides with the first Chern class of the complexification. The Pontryagin classes p_k(E) \in H^{4k}(B; \mathbb{Z}) for real bundles of rank at least $4k are derived from Chern classes via p_k(E) = (-1)^k c_{2k}(E \otimes \mathbb{C}), linking real and complex invariants and satisfying analogous axioms. Examples illustrate these classes' utility: for the tangent bundle TS^2 of the 2-sphere, viewed as a complex line bundle, c_1(TS^2) = 2g, where g \in H^2(S^2; \mathbb{Z}) is the positive generator, reflecting the bundle's non-triviality. The hairy ball theorem follows from the Euler class, as e(TS^{2m}) \neq 0 for even-dimensional spheres, implying no nowhere-vanishing vector field.

Vector Bundles in K-Theory

Topological K-theory associates to a compact Hausdorff space B the abelian group K^0(B), defined as the Grothendieck group of the semigroup of isomorphism classes of complex vector bundles over B under direct sum, where elements are formal differences [E] - [F] of bundle classes and relations arise from short exact sequences $0 \to F \to E \oplus H \to G \to 0 yielding [E] = [F] + [G].^[31] This construction incorporates stable equivalence, where bundles E and F are stably isomorphic if E \oplus H \cong F \oplus H for some bundle H.^[1] The group K^0(B) acquires a ring structure from the tensor product of bundles, with the trivial line bundle as the unit.^[31] For connected B, the reduced group \tilde{K}^0(B) is the kernel of the rank map K^0(B) \to \mathbb{Z} = K^0(\mathrm{pt}), capturing the non-trivial stable classes orthogonal to the trivial bundle.^[1] Bott periodicity asserts that the higher K-groups, defined via suspension \Sigma B = S^1 \wedge B, satisfy K^{n+2}(B) \cong K^n(B) for all n \in \mathbb{Z}, endowing K-theory with a \mathbb{Z}/2\mathbb{Z}-periodic cohomology theory.^[32] K-theory is contravariant: for a continuous map f: B' \to B, the pullback induces a ring homomorphism f^*: K^0(B) \to K^0(B') by f^*[E] = [f^*E].^[31] For a CW pair (X, A) with inclusion i: A \hookrightarrow X and quotient p: X \to X/A, the long exact sequence is

\cdots \to \tilde{K}^0(A) \xrightarrow{i^*} \tilde{K}^0(X) \xrightarrow{p^*} \tilde{K}^0(X/A) \xrightarrow{\partial} \tilde{K}^{-1}(A) \to \cdots,

where the boundary map \partial detects extensions of bundles.^[1] The Chern character provides a natural ring homomorphism \mathrm{ch}: K^0(B) \to H^*(B; \mathbb{Q}) from K-theory to rational cohomology, expressing bundle classes in terms of their Chern classes via the formula

\mathrm{ch}(E) = \mathrm{rank}(E) + c_1(E) + \frac{c_1(E)^2 - 2c_2(E)}{2!} + \cdots + \frac{1}{k!} \sum \sigma_k(c_1(E), \dots, c_k(E)) + \cdots,

where \sigma_k denotes the k-th power sum of the formal Chern roots, thus linking stable bundle invariants to topological cohomology.^[33] A representative example is K^0(S^2) \cong \mathbb{Z} \oplus \mathbb{Z}, where the first factor is generated by the trivial bundle (via rank) and the reduced group \tilde{K}^0(S^2) \cong \mathbb{Z} by the Hopf line bundle \mathcal{O}(1) over \mathbb{CP}^1 \cong S^2.^[1] Vector bundles over S^2 are classified up to stable isomorphism by clutching constructions: a bundle is obtained by gluing trivial bundles over the hemispheres via a transition function S^1 \to \mathrm{U}(n), and the stable classes correspond to homotopy classes [S^1, U(n)], yielding \mathbb{Z} from the determinant line bundle's winding number.^[1]

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