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Pullback bundle

In mathematics, particularly within algebraic topology and differential geometry, a pullback bundle, also known as an induced bundle, is a fiber bundle f^*E over a base space X that is constructed from a given fiber bundle \pi: E \to B with structure group G and fiber F, via a continuous map f: X \to B.^[1]^[2] The total space of the pullback bundle is defined as the subset f^*E = \{(x, e) \in X \times E \mid f(x) = \pi(e)\}, with the projection f^*\pi: f^*E \to X given by (x, e) \mapsto x, ensuring that the fiber over each x \in X is canonically identified with the fiber E_{f(x)} of the original bundle.^[1]^[2] The construction of a pullback bundle preserves the bundle structure: if the original bundle has local trivializations over an open cover \{U_\alpha\} of B with transition functions g_{\alpha\beta}: U_\alpha \cap U_\beta \to G, then the pullback inherits trivializations over \{f^{-1}(U_\alpha)\} with transition functions f^*g_{\alpha\beta}(x) = g_{\alpha\beta}(f(x)), confirming that f^*E is indeed a fiber bundle with the same structure group G and fiber F.^[2] This yields a bundle map (f, \hat{f}): f^*E \to E, where \hat{f}(x, e) = e, forming a commutative diagram with the projections \pi and f^*\pi.^[2] Moreover, there is a natural isomorphism between pullback bundles induced by homotopic maps; if f \simeq g: X \to B are homotopic, then f^*E \cong g^*E as bundles over X.^[1]^[2] Pullback bundles play a central role in the study of fiber bundles, enabling the transfer of bundle-theoretic constructions across spaces and facilitating the computation of invariants such as characteristic classes.^[1] For instance, the pullback of a trivial bundle is always trivial, underscoring the functorial nature of the construction in the category of fiber bundles.^[2] They are particularly useful in applications like defining sections over submanifolds, inducing connections and metrics from the original bundle, and analyzing homotopy equivalences in topological contexts.^[1]^[2]

Fundamentals of Fiber Bundles

Definition and Basic Structure

A fiber bundle is formally defined as a triple (E, \pi, B), consisting of a total space E, a base space B, and a continuous surjective projection map \pi: E \to B.^[3] The base space B is typically a topological space, such as a manifold, while the total space E is equipped with the topology that makes \pi a surjective map with certain local product properties.^[4] For each point b \in B, the fiber over b is the preimage \pi^{-1}(b), which is required to be homeomorphic to a fixed topological space F, known as the typical fiber.^[3] This homeomorphism ensures that all fibers are structurally identical, though the global twisting of the bundle may prevent a global product structure.^[4] The bundle structure is locally trivial, meaning that around every point in B, there exists a neighborhood that is homeomorphic to a product of that neighborhood with the fiber F, compatible with the projection \pi.^[3] Associated with the fiber bundle is a structure group G, which is a topological group acting continuously on the fiber F from the left.^[3] The local trivializations are required to be G-equivariant, meaning that the homeomorphisms respect the group action, and bundle maps between fiber bundles with the same fiber and structure group are also G-equivariant.^[4] This framework allows for the classification of bundles up to isomorphism via the action of G. The concept of fiber bundles was formalized in the late 1930s and early 1940s, with key contributions from Hassler Whitney, who developed the theory in the context of sphere bundles and characteristic classes during 1935–1941.^[5] Whitney's work laid the groundwork for modern bundle theory in algebraic topology. A standard example is the trivial bundle B \times F \to B, where the projection is the natural one onto the first factor, and the fibers are canonically identified with F without any twisting.^[3] This serves as the simplest case, illustrating the product structure that more general bundles locally resemble.^[4]

Local Trivializations

A fiber bundle (\pi: E \to B, F) is equipped with local trivializations via an open cover \{U_i\}_{i \in I} of the base space B and continuous fiber-preserving homeomorphisms \phi_i: \pi^{-1}(U_i) \to U_i \times F for each i, satisfying \pi \circ \phi_i = \mathrm{pr}_1, where \mathrm{pr}_1: U_i \times F \to U_i is the canonical projection onto the first factor.^[6] These homeomorphisms ensure that the bundle restricts locally over each U_i to the trivial product bundle U_i \times F.^[7] On pairwise overlaps U_i \cap U_j, the trivializations \phi_i and \phi_j are compatible via continuous transition functions g_{ij}: U_i \cap U_j \to [G](/page/G), where G is the structure group acting effectively and continuously on the typical fiber F (often taken as the group of homeomorphisms \mathrm{Homeo}(F)). Specifically, for x \in U_i \cap U_j and f \in F,

\phi_j \circ \phi_i^{-1}(x, f) = (x, g_{ij}(x) \cdot f),

where \cdot denotes the group action.^[6]^[7] These functions encode how the local product structures glue together globally. The transition functions further satisfy the cocycle condition on triple overlaps U_i \cap U_j \cap U_k: for all x in the intersection,

g_{ik}(x) = g_{ij}(x) \cdot g_{jk}(x),

ensuring consistency in the gluing and that the resulting space E is well-defined as a topological space.^[6]^[7] A concrete example arises with the Möbius line bundle over S^1, a non-orientable real vector bundle of rank 1 with fiber F = \mathbb{R}. It admits trivializations over the open cover \{U_0, U_1\}, where U_0 = S^1 \setminus \{1\} and U_1 = S^1 \setminus \{-1\}, and the sole nontrivial transition function is the constant map g_{01}: U_0 \cap U_1 \to \{\pm 1\} \subseteq \mathrm{GL}(1, \mathbb{R}) given by g_{01}(x) = -1, which introduces the characteristic twist.^[7]

Constructing the Pullback Bundle

Formal Definition via Pullback Construction

Given a fiber bundle p: E \to B with typical fiber F and a continuous map f: B' \to B between base spaces, the pullback bundle, denoted f^* E or E', is constructed as the fiber product

E' = \{ (b', e) \in B' \times E \mid f(b') = p(e) \},

equipped with the projection p': E' \to B' given by the first coordinate, p'(b', e) = b'.^[8] This subspace of the product space inherits the subspace topology from B' \times E, ensuring p' is continuous.^[8] The fiber of p' over a point b' \in B' is (p')^{-1}(b') = \{ b' \} \times p^{-1}(f(b')), which is homeomorphic to the original fiber F_{f(b')} over f(b') via the second coordinate projection.^[8] If the original bundle E has structure group G acting on F, the pullback E' inherits the same G-action on its fibers, preserving the bundle's structure.^[9] Category-theoretically, the pullback bundle f^* E is the limit of the diagram B' \xrightarrow{f} B \xleftarrow{p} E in the category of fiber bundles, satisfying a universal property: for any fiber bundle E'' \to B' equipped with a bundle map \phi: E'' \to E over f (i.e., p \circ \phi = f \circ p''), there exists a unique bundle map \psi: E'' \to f^* E over the identity on B' such that the composite E'' \xrightarrow{\psi} f^* E \to E equals \phi.^[10] This construction ensures the commutative diagram

\begin{CD} E' @>{\mathrm{pr}_2}>> E \\ @V{p'}VV @VV{p}V \\ B' @>>f> B, \end{CD}

where \mathrm{pr}_2 is the second coordinate projection restricted to E', reflecting that p \circ \mathrm{pr}_2 = f \circ p'.^[8]

Verification as a Fiber Bundle

To verify that the pullback construction yields a valid fiber bundle, consider a fiber bundle \pi: E \to B with fiber F and a continuous map f: B' \to B. The pullback bundle is defined with total space E' = \{(b', e) \in B' \times E \mid f(b') = \pi(e)\} and projection p': E' \to B' given by p'(b', e) = b'.^[7]^[11] Suppose the original bundle admits an open cover \{U_i\}_{i \in I} of B with local trivializations \phi_i: \pi^{-1}(U_i) \to U_i \times F. The pullback inherits a cover \{f^{-1}(U_i)\}_{i \in I} of B', over each of which it is trivialized by the map \phi_i': p'^{-1}(f^{-1}(U_i)) \to f^{-1}(U_i) \times F defined by (b', e) \mapsto (b', \pr_2 (\phi_i (e))), where \pr_2 : U_i \times F \to F denotes the projection onto the second factor. This construction ensures that \phi_i' is a homeomorphism, as it composes the original trivialization with the identification induced by f, preserving the product structure locally.^[7]^[11] The transition functions of the pullback bundle are induced from those of the original: if g_{ij}: U_i \cap U_j \to \mathrm{Homeo}(F) are the transition maps for \phi_i and \phi_j, then the pullback transition maps are \tilde{g}_{ij} = g_{ij} \circ f: f^{-1}(U_i \cap U_j) \to \mathrm{Homeo}(F). These satisfy the cocycle condition \tilde{g}_{ik} = \tilde{g}_{ij} \circ \tilde{g}_{jk} on triple overlaps f^{-1}(U_i \cap U_j \cap U_k), since the original g_{ij} do and composition with the continuous map f preserves this relation: (g_{ij} \circ f) \circ (g_{jk} \circ f) = g_{ik} \circ f. Thus, the local trivializations glue consistently to define a fiber bundle structure on E'.^[7] A sketch of the verification proceeds by confirming the fiber bundle axioms. Over each b' \in B', the fiber p'^{-1}(b') consists of pairs (b', e) with \pi(e) = f(b'), which is homeomorphic to the original fiber \pi^{-1}(f(b')) via the map e \mapsto (b', e). This homeomorphism extends locally over neighborhoods, and since the original local trivializations are homeomorphisms to product spaces, the pulled-back maps \phi_i' preserve the bundle structure, ensuring p' is a locally trivial fibration with fiber F.^[7]^[11] For smooth fiber bundles, such as vector bundles over manifolds, the pullback inherits a differentiable structure when f is smooth. The local trivializations \phi_i' become diffeomorphisms if the original \phi_i are, as the composition with f (a diffeomorphism locally) and projections preserves smoothness, yielding a smooth projection p'.^[7] A representative example is the pullback of the tangent bundle TM \to M along a smooth embedding \iota: N \to M: the resulting bundle \iota^* TM \to N is a smooth vector bundle over N, with fibers T_{\iota(n)} M and local trivializations induced by those of TM composed with \iota.^[7]^[11]

Core Properties

Existence and Uniqueness

The existence of the pullback bundle f^*E for a fiber bundle p: E \to B and a continuous map f: B' \to B is guaranteed in the category of topological spaces, where the total space is defined as E' = \{ (b', e) \in B' \times E \mid f(b') = p(e) \} with projection p': E' \to B' given by p'(b', e) = b'.^[7] The projection p' forms a locally trivial fiber bundle with the same fiber as p, as local trivializations over open covers of B pull back to trivializations over the preimages in B'.^[6] This construction satisfies the fiber bundle axioms, ensuring existence as a topological bundle; this holds in the general topological setting, though Hausdorff paracompactness of the bases is often assumed in applications such as differential geometry.^[2] Uniqueness holds up to bundle isomorphism over the identity map on B', stemming from the universal property of the pullback: any bundle morphism \eta: E'' \to E over f (i.e., commuting with the projections via f) factors uniquely through f^*E via a bundle isomorphism over \mathrm{id}_{B'}.^[7] This isomorphism is constructed by leveraging the local trivializations, ensuring that the pullback is the terminal object in the category of bundles over B' mapping to E over f.^[6] Consequently, any two constructions satisfying the pullback diagram are canonically isomorphic, preserving the bundle structure without ambiguity.^[12] In the special case of principal G-bundles, the pullback preserves the principal structure exactly: if p: P \to B is a principal G-bundle, then p': f^*P \to B' admits a free and transitive right G-action defined by (b', e) \cdot g = (b', e \cdot g), making f^*P a principal G-bundle isomorphic over f to the original via the canonical projection.^[7] This preservation follows directly from the right-invariance of the bundle's transition functions under the group action.^[12] An isomorphism criterion for pullback bundles states that two such bundles over the same f are isomorphic if and only if their transition functions agree after composition with f: specifically, for local trivializations over covers \{U_i\} of B and \{V_j = f^{-1}(U_j)\} of B', the transitions g'_{jk}: V_j \cap V_k \to G satisfy g'_{jk} = g_{jk} \circ f.^[7] This condition ensures the bundle maps induced by the transitions are compatible, yielding a fiber-preserving homeomorphism between the total spaces.^[2]

Functoriality and Natural Transformations

The pullback operation defines a contravariant functor f^*: \mathbf{Bund}(B) \to \mathbf{Bund}(B') between the categories of fiber bundles over bases B and B', where a continuous map f: B' \to B induces, on objects, the pullback bundle f^* E for any bundle E \to B, and on morphisms, the bundle map f^* \phi: f^* E \to f^* E' for a bundle morphism \phi: E \to E' over \mathrm{id}_B.^[6] This functoriality arises because the pullback construction respects the categorical structure, mapping bundle projections and compatible maps accordingly, while reversing the direction of base maps to reflect the contravariant nature.^[13] For composable base maps g: B'' \to B' and f: B' \to B, the pullback satisfies the isomorphism (g \circ f)^* E \cong g^* (f^* E) as bundles over B'', where the explicit isomorphism is induced by the iterated fiber product: points in (g \circ f)^* E correspond to triples (b'', e) with (g \circ f)(b'') = \pi_E(e), which via f^* E yield pairs (b', e) with f(b') = \pi_E(e) and g(b'') = b', matching the structure of g^* (f^* E).^[13] This compatibility ensures that pullback preserves the composition of base morphisms up to canonical isomorphism, reinforcing its functorial behavior.^[6] The identity map \mathrm{id}_B: B \to B induces the identity functor on \mathbf{Bund}(B), via the natural isomorphism \mathrm{id}_B^* E \cong E given by the evident projection and inclusion between E and the fiber product B \times_E E.^[13] More broadly, the pullback functor relates contravariantly to the direct image (pushforward) functor, which is covariant; together, they form an adjoint pair in appropriate settings, such as for vector bundles over manifolds, where f^* \dashv f_*.^[6] In the subcategory of vector bundles, the pullback functor preserves key operations, such as rank (with \mathrm{rank}(f^* E) = \mathrm{rank}(E)) and tensor products, via the natural isomorphism f^*(E \otimes F) \cong f^* E \otimes f^* F over B', constructed fiberwise from the universal bilinear map on vector spaces.^[14] This compatibility extends to other algebraic structures, like duals and exterior powers, making pullback essential for transferring bundle constructions across base spaces. The universal property of the pullback bundle characterizes it uniquely up to isomorphism: given any bundle Q \to B' and a bundle map \psi: Q \to E over f (i.e., \pi_E \circ \psi = f \circ \pi_Q), there exists a unique bundle map \tilde{\psi}: Q \to f^* E over \mathrm{id}_{B'} (i.e., \pi_{f^* E} \circ \tilde{\psi} = \pi_Q) such that the diagram commutes:

\begin{tikzcd} Q \arrow[r, "\tilde{\psi}"] \arrow[d, "\pi_Q"'] & f^* E \arrow[d, "\pi_{f^* E}"] \arrow[l, "\psi"'] \\ B' \arrow[r, "f"] & B \end{tikzcd}

with \tilde{\psi}(q) = ( \pi_Q(q), \psi(q) ) for q \in Q, ensuring the fiber product captures all compatible lifts.^[13] This property underpins the functoriality, as it guarantees that pullback maps are well-defined and unique for morphisms in \mathbf{Bund}(B).^[6]

Applications and Relations

In Differential Geometry

In differential geometry, the pullback construction plays a central role in transferring geometric structures from one manifold to another via smooth maps. For a smooth map f: N \to M between manifolds and a vector bundle E \to M, the pullback bundle f^* E \to N consists of pairs (p, e) \in N \times E such that f(p) = \pi_M(e), where \pi_M: E \to M is the projection, equipped with the projection \pi_N: f^* E \to N given by \pi_N(p, e) = p. This bundle is itself a vector bundle over N, with fibers isomorphic to those of E. A key example arises with the tangent bundle TM \to M: if f: N \to M is an immersion, then f^* TM \to N represents the ambient tangent bundle restricted to the image of N, allowing the study of tangent spaces along the immersed submanifold without extending sections arbitrarily.^[15] This pullback induces compatible structures on associated geometric objects. If E \to M is equipped with a connection \nabla, the pullback bundle f^* E inherits an induced connection f^* \nabla defined by (f^* \nabla)_X (f^* s) = f^* (\nabla_{df(X)} s) for vector fields X on N and sections s of E, where f^* s denotes the pulled-back section. This ensures that parallel transport and covariant derivatives are preserved under the map f, facilitating computations of curvature and holonomy on the source manifold. Similarly, for a Riemannian metric g on M, the pullback f^* g on N is defined by (f^* g)_p(u, v) = g_{f(p)}(df_p(u), df_p(v)) for u, v \in T_p N, yielding a Riemannian metric on N when f is an immersion. This induced metric is particularly useful for submanifolds, where it restricts the ambient metric to the tangent spaces, enabling analysis of extrinsic geometry such as mean curvature and second fundamental forms.^[16]^[17] An important application appears in general relativity, where the spacetime manifold (M, g) carries a Lorentzian metric g. Along a worldline \gamma: I \to M parameterized by proper time, the pullback \gamma^* g restricts the spacetime metric to the curve, yielding the line element ds^2 = -d\tau^2 for timelike paths, which defines proper time intervals and four-velocities essential for particle dynamics and observer measurements. Pullbacks also preserve additional structures like orientations and spin structures under suitable conditions. If E \to M is an oriented vector bundle and f: N \to M is an orientation-preserving diffeomorphism, then f^* E inherits the orientation via the pullback of transition functions, which remain orientation-preserving. For spin structures, which refine orientations to lift to the spin group, the pullback f^* S of a spin structure S on E exists and is compatible if f preserves the orientation, ensuring consistency in spinor fields and Dirac operators on the pulled-back bundle.^[18]^[19]^[20]

Connection to Sheaf Theory

In the context of fiber bundles, the sheaf of sections provides a sheaf-theoretic perspective on the bundle structure. For a fiber bundle E \to B with projection \pi, the sheaf of smooth sections \Gamma(E) on B assigns to each open set U \subset B the space of smooth sections over U, and its stalk at a point b \in B is isomorphic to the fiber E_b.^[21] This construction endows the bundle with a local structure that aligns with the sheaf's gluing properties, ensuring compatibility with the bundle's topology.^[22] The pullback operation extends naturally to sheaves of sections. Given a smooth map f: B' \to B and the sheaf \Gamma(E) on B, the inverse image sheaf f^{-1} \Gamma(E) on B' has sections over an open U' \subset B' given by \Gamma(f(U'), E), the smooth sections of E over f(U').^[23] This sheaf is naturally isomorphic to the sheaf of sections \Gamma(f^* E) of the pullback bundle f^* E \to B', preserving the local triviality and transition functions of the original bundle.^[23] In algebraic geometry, this correspondence deepens for quasi-coherent sheaves associated to vector bundles. The pullback of a quasi-coherent sheaf \tilde{M} along a morphism f: Y \to X yields f^* \tilde{M} = \tilde{M \otimes_R S}, where X = \mathrm{Spec}(R) and Y = \mathrm{Spec}(S), which corresponds directly to the pullback of the associated vector bundle.^[24] For line bundles, this identifies the pullback sheaf with the induced line bundle, facilitating computations in projective schemes.^[24] Étale pullbacks further highlight the sheaf perspective by preserving exactness in sequences. For an étale morphism \pi: Y \to X, the pullback functor \pi^* on sheaves over the étale site is exact, mapping short exact sequences of sheaves on X_{\mathrm{\acute{e}t}} to short exact sequences on Y_{\mathrm{\acute{e}t}}.^[25] In contrast, the topological pullback does not generally preserve exactness, as it lacks the étale condition's rigidity on fiber variations.^[25] While pullback bundles require constant fiber structure, sheaf pullbacks accommodate varying fibers, allowing more general data attachment. However, for constant sheaves—such as the constant sheaf with stalk a fixed abelian group—the pullback coincides with the sections of the trivial bundle pullback, recovering the bundle case via the associated constant fiber bundle.^[23] Historically, Ehresmann's work in the 1950s on connections and fibrations laid groundwork for linking bundle classifications to sheaf cohomology, with Serre's sheaf-theoretic developments enabling cohomology groups like H^1(B, \Gamma(E)) to classify bundles up to isomorphism.^[26]

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