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Associated bundle

In mathematics, particularly in the fields of algebraic topology and differential geometry, an associated bundle is a fiber bundle constructed from a principal bundle and a representation of its structure group on a suitable space, allowing the transfer of geometric structures from the principal bundle to a new fiber type.^[1] Specifically, given a principal G-bundle \pi: P \to M over a base manifold M and a left G-action on a space F (such as a vector space V via a linear representation \rho: G \to \mathrm{GL}(V)), the associated bundle has total space P \times_G F = (P \times F)/\sim, where the equivalence relation identifies (p \cdot g, f) \sim (p, g \cdot f) for g \in G, and projection map \hat{\pi}([p, f]) = \pi(p) to M, yielding fibers diffeomorphic to F.^[2] This construction preserves local triviality and equivariance, ensuring the associated bundle inherits the topological and smooth structure of the original principal bundle while adapting the fiber to model phenomena like tangent spaces or gauge fields.^[3] Associated bundles play a central role in unifying various types of fiber bundles under a common framework, particularly when the action on F is effective, establishing an equivalence between principal bundles and their associated fiber bundles up to isomorphism.^[4] For instance, the tangent bundle of a Riemannian manifold arises as the associated vector bundle to its orthogonal frame bundle via the standard representation of the orthogonal group on \mathbb{R}^n.^[1] In broader contexts, such as gauge theory, associated bundles facilitate the description of matter fields transforming under group representations, with connections on the principal bundle inducing compatible structures on the associated bundle.^[2] The theory extends to higher categorical settings, where associated bundles can be viewed as homotopy pullbacks in groupoid models, emphasizing their role in modern geometric and topological applications.^[5]

Definitions and Motivation

Definition

In the context of differential geometry and topology, a fiber bundle is a structure consisting of a total space E, a base space M, a projection map \pi: E \to M, and fibers diffeomorphic to a fixed space F, with local trivializations ensuring that the bundle is locally equivalent to the product U \times F for open sets U \subset M.^[3] A principal G-bundle, where G is a Lie group serving as the structure group, refines this by taking the fiber to be G itself, equipped with a free and transitive right G-action P \times G \to P, (p, g) \mapsto p \cdot g, such that the projection \pi: P \to M is constant on G-orbits and local sections exist over coordinate charts, inducing local trivializations P|_U \cong U \times G.^[3]^[2] Given such a principal G-bundle P \to M and a representation \rho: G \to \mathrm{GL}(V) of G on a vector space V, which defines a left G-action on V via g \cdot v = \rho(g) v, the associated bundle is the fiber bundle E = P \times_\rho V \to M with typical fiber V.^[2] The total space E carries the quotient topology from the product P \times V, and the structure group G acts on the fibers through \rho. The projection \pi_E: E \to M is defined by \pi_E([p, v]) = \pi_P(p), where [\cdot, \cdot] denotes equivalence classes.^[3] The explicit construction proceeds via the equivalence relation on P \times V: (p \cdot g, v) \sim (p, \rho(g^{-1}) v) for all p \in P, v \in V, and g \in G, ensuring the G-action is compatible with the right action on P.^[2] Thus, E = (P \times V)/{\sim}, and each fiber \pi_E^{-1}(m) \cong V inherits a vector space structure from V, making E a vector bundle when V is finite-dimensional. Local sections of P lift to trivializations of E, confirming its fiber bundle structure.^[3]

Historical Motivation

The concept of associated bundles emerged in the mid-20th century as part of the rapid development of fiber bundle theory in algebraic topology, driven by the need to classify complex geometric structures and understand their homotopy properties. Norman Steenrod played a pivotal role in formalizing this idea in his 1951 monograph The Topology of Fibre Bundles, where he introduced associated bundles in Section 9 as a construction allowing new fiber bundles to be derived from a principal bundle via a representation of the structure group on a fiber space.^[6] This innovation was motivated by the desire to unify disparate types of bundles—such as vector bundles and principal bundles—under a common framework, enabling systematic classification through classifying spaces and addressing questions about bundle equivalence and obstructions to sections.^[4] In the 1950s, Jean-Pierre Serre extended these ideas into algebraic geometry, establishing a correspondence between vector bundles and finitely generated projective modules over the structure sheaf in his seminal 1955 paper "Faisceaux algébriques cohérents," motivated by the goal of computing cohomology and resolving Riemann-Roch-type problems on algebraic varieties.^[7] This work highlighted the role of vector bundles in bridging topology and algebra, facilitating the study of characteristic classes and stable equivalence of bundles over schemes.^[8] Concurrently, Shiing-Shen Chern's development of characteristic classes for complex vector bundles provided further impetus for bundle theory, as these classes captured global topological invariants like Euler and Chern numbers. Chern's 1946 paper "Characteristic Classes of Hermitian Manifolds" laid the groundwork by associating differential forms to bundle curvatures, motivated by the need to generalize Stiefel-Whitney classes to complex settings and quantify obstructions in differential geometry.^[9] By the late 1950s, this formalism influenced the study of G-structures and reductions of structure groups, where associated bundles served to model transformed fibers under symmetry actions.^[10] The introduction of associated bundles also found retrospective motivation in physics through gauge theories, where principal bundles represent symmetry groups and associated vector bundles encode matter fields transforming under those symmetries, such as fermions in representation spaces. This connection became explicit in the 1970s with the geometric formulation of Yang-Mills theories, but the underlying bundle constructions from the 1950s provided the mathematical foundation for interpreting gauge fields as connections on principal bundles and their associated counterparts.^[11]

Constructions

From Principal Bundles

The construction of an associated bundle begins with a principal G-bundle P \to M, where G is a topological group acting freely and properly on the right on the total space P, and M is the base space. To form the associated bundle with fiber type F, select a left action of G on F, which can be given by a representation \rho: G \to \Aut(F). The total space E is then obtained as the quotient E = (P \times F)/G, where the group acts diagonally via (p, f) \cdot g = (p g, \rho(g^{-1}) f) for p \in P, f \in F, and g \in G. This equivalence relation identifies points under the group action, ensuring the quotient is well-defined due to the free action on P.^[12]^[3] The topology on E is the quotient topology induced from the product topology on P \times F. Local trivializations are inherited from those of the principal bundle P: for an open set U \subset M over which P|_U \cong U \times [G](/page/G) via a bundle trivialization \phi_U: \pi^{-1}(U) \to U \times [G](/page/G), the restricted associated bundle satisfies E|_U \cong U \times F, where the isomorphism sends the equivalence class [p, f] (with \phi_U(p) = (x, g)) to (x, \rho(g) f). This ensures E \to M is a fiber bundle with structure group G and typical fiber F.^[12]^[3] In the special case where F = V is a vector space and \rho: G \to \GL(V) is a linear representation, the resulting bundle E \to M is a vector bundle of rank \dim V with G-structure, meaning the transition functions take values in the image of \rho. The vector space operations on V descend to the fibers of E, making addition and scalar multiplication well-defined and smooth.^[12]^[3]^[2] The projection \pi: E \to M defined by \pi([p, f]) = \pi_P(p), where \pi_P: P \to M is the projection of the principal bundle, is continuous and smooth when P, M, and G are smooth manifolds with G a Lie group. Each fiber \pi^{-1}(m) is diffeomorphic to F via the map sending [p, f] to f for any p \in \pi_P^{-1}(m), as the action restricts to a free transitive action on the fiber copies. This diffeomorphism is independent of the choice of representative due to the equivalence relation.^[3]^[2]

From Fiber Bundles

In the context of fiber bundles, the construction of an associated principal bundle proceeds by identifying the space of "frames" within each fiber, which captures the action of the structure group. Consider a fiber bundle \pi: E \to M with typical fiber F and structure group G, where G acts effectively (and smoothly) on F from the left. The frame bundle P(E), or associated principal G-bundle, is defined as the total space whose fiber over each x \in M consists of all G-equivariant diffeomorphisms \phi: F \to E_x, i.e., smooth bijections satisfying \phi(g \cdot f) = g \cdot \phi(f) for all g \in G and f \in F, with the induced action on E_x inherited from the bundle structure via local trivializations. The projection P(E) \to M sends each frame \phi over x to x, and the right G-action on P(E) is given by post-composition: (\phi \cdot g)(f) = \phi(g^{-1} \cdot f) for g \in G, which is free and transitive on each fiber provided the original action of G on F is effective and the bundle is locally trivial.^[13] This construction ensures that P(E) is a principal G-bundle because local trivializations of E over open covers \{U_i\} of M induce G-equivariant trivializations of P(E) over the same covers, with transition functions matching those of E. Specifically, if \psi_i: \pi^{-1}(U_i) \to U_i \times F is a local trivialization for E, then a frame \phi over x \in U_i corresponds to the constant map sending F to the standard fiber via \psi_i, and overlaps yield transition maps in G acting on frames by right multiplication. The local triviality of E thus guarantees that P(E) inherits the principal bundle structure, with fibers diffeomorphic to G under the assumption that G acts freely and transitively on the set of frames (which holds when the action on F is effective and F is a homogeneous G-space). The original fiber bundle E recovers as the associated bundle to this frame bundle P(E) via the tautological representation \rho: G \to \mathrm{Aut}(F), where \rho(g) is the standard left action of g on F. Explicitly, E \cong P(E) \times_\rho F, where the quotient identifies [ \phi, f ] \sim [ \phi \cdot g, \rho(g^{-1})(f) ] = [ \phi \cdot g, g^{-1} \cdot f ] for \phi \in P(E)_x, f \in F, and g \in G; the projection to M is well-defined, and fibers are diffeomorphic to F with the induced G-action. This isomorphism is canonical, preserving the bundle structure and equivariance.^[14] Under these conditions, every fiber bundle with fiber F and structure group G (acting effectively) arises uniquely up to isomorphism as the associated bundle to its frame bundle P(E). The uniqueness follows from the fact that any two such frame bundles over the same base are isomorphic as principal G-bundles if and only if the original fiber bundles are isomorphic, as the frames determine the equivariant identifications of fibers. This reversibility underscores the duality between fiber bundles and principal bundles in the category of bundles with fixed fiber and structure group.^[13]

Examples

Basic Topological Examples

The simplest example of an associated bundle arises when the base space M is a single point. In this case, the principal [G](/page/G)-bundle P \to M is simply the discrete group G itself as the total space, with the projection mapping every element to the point. For a left [G](/page/G)-action \rho on a space [V](/page/V.), the associated bundle is E = P \times_\rho V = (G \times V)/\sim, where (g, v) \sim (g', v') if there exists h \in G such that g' = g h and v' = \rho(h^{-1}) v. This quotient is homeomorphic to the orbit space V/G, where G acts via \rho. A fundamental non-trivial topological example is provided by the Hopf fibration, the principal S^1-bundle \pi: S^3 \to S^2 given by [z_0, z_1] \mapsto [z_0 : z_1] in homogeneous coordinates, where S^3 \subset \mathbb{C}^2 is the unit sphere and the base S^2 is identified with \mathbb{C}P^1. The associated complex line bundle, using the standard representation \rho: S^1 \to U(1) on \mathbb{C}, is the tautological line bundle over \mathbb{C}P^1, whose total space consists of pairs (L, v) where L \in \mathbb{C}P^1 is a line in \mathbb{C}^2 and v \in L. This bundle is constructed via clutching functions over the northern and southern hemispheres of S^2, with transition function f(z) = z for z \in S^1 on the equator.^[15] Another basic example is the non-trivial real line bundle over the circle S^1, whose total space is the open Möbius strip. This arises as the associated bundle to the non-trivial principal \mathbb{Z}/2\mathbb{Z}-bundle over S^1, which is the double covering S^1 \to S^1 given by z \mapsto z^2 (with \mathbb{Z}/2\mathbb{Z} = \{1, -1\} acting by z \sim -z). The representation is the sign action \rho: \mathbb{Z}/2\mathbb{Z} \to \mathrm{GL}(1, \mathbb{R}) on the fiber \mathbb{R}, where \rho(1)v = v and \rho(-1)v = -v. The total space is the quotient (S^1 \times \mathbb{R})/\sim, where (z, v) \sim (-z, -v), yielding the twisted line bundle that captures non-orientability. This construction illustrates how associated bundles encode topological twists via group actions in low dimensions.^[16] In general, associated bundles over spheres S^n are classified using clutching functions derived from the homotopy groups of the structure group. For a principal [G](/page/G)-bundle over S^n, the isomorphism classes correspond to elements of \pi_{n-1}([G](/page/G)), where the clutching function f: S^{n-1} \to [G](/page/G) glues the trivial bundles over the upper and lower hemispheres D^n_+ and D^n_- via right multiplication by f. The associated bundle E then inherits this classification, with transition functions \rho(f): V \to V on the fibers, providing a concrete way to enumerate bundles in low dimensions, such as line bundles over S^2 via \pi_1([G](/page/G)).^[15]

Geometric and Gauge Examples

In differential geometry, the tangent bundle of a smooth manifold provides a fundamental example of an associated bundle. Consider an n-dimensional manifold M; its frame bundle P(M) is the principal \mathrm{GL}(n,\mathbb{R})-bundle over M whose fibers consist of ordered bases of tangent spaces. The tangent bundle TM is then the associated bundle obtained via the standard representation of \mathrm{GL}(n,\mathbb{R}) on \mathbb{R}^n, given explicitly by TM = P(M) \times_{\mathrm{GL}(n,\mathbb{R})} \mathbb{R}^n, where the equivalence relation identifies (p, v) \sim (p g, g^{-1} v) for g \in \mathrm{GL}(n,\mathbb{R}) and v \in \mathbb{R}^n.^[17] This construction equips TM with a canonical vector bundle structure, facilitating the study of tangent vectors as sections and enabling the definition of connections on M. In gauge theory, particularly Yang-Mills theory, associated bundles arise naturally in the description of gauge fields and their curvatures. For a principal G-bundle P over a spacetime manifold (typically a Lorentzian 4-manifold), the adjoint bundle \mathrm{ad}(P) = P \times_G \mathfrak{g} is formed using the adjoint representation of G on its Lie algebra \mathfrak{g}, where (p, X) \sim (p g, \mathrm{Ad}_g X) for g \in G and X \in \mathfrak{g}. A connection on P is a \mathfrak{g}-valued 1-form, and its curvature 2-form takes values in sections of \mathrm{ad}(P), encoding the field strength in Yang-Mills equations.^[18] This setup underlies non-Abelian gauge interactions, where matter fields may transform under further associated bundles via other representations of G.^[14] Spinor bundles exemplify associated bundles in the context of structure group reductions, essential for Dirac operators and fermionic fields in curved spacetime. Starting from the orthonormal frame bundle with structure group \mathrm{SO}(n), a spin structure reduces it to a principal \mathrm{Spin}(n)-bundle P_{\mathrm{Spin}} over the manifold; the spinor bundle S is then the associated bundle S = P_{\mathrm{Spin}} \times_{\mathrm{Spin}(n)} \Delta, where \Delta is the spinor representation space (a module over the Clifford algebra \mathrm{Cl}_n).^[19] Over Lorentzian manifolds, such as 4-dimensional spacetime, this yields chiral spinor bundles for Weyl fermions, with the reduction enabling the double cover \mathrm{Spin}(1,3) \to \mathrm{SO}(1,3) to resolve sign ambiguities in spinor transformations.^[20] Sections of S represent spinor fields, crucial for coupling to gravitational and gauge connections. Instantons illustrate associated bundles in the study of self-dual connections, bridging geometry and quantum field theory. On \mathbb{R}^4 (or compactified S^4), consider a principal \mathrm{SU}(2)-bundle P; self-dual instantons are connections whose curvature F_A satisfies *F_A = F_A, where * is the Hodge star. Associated vector bundles E = P \times_{\mathrm{SU}(2)} V arise via representations on finite-dimensional complex vector spaces V, such as the fundamental 2-dimensional representation, yielding rank-2 bundles with Chern class c_2(E) = k for instanton number k.^[21] The BPST instanton, the simplest non-trivial example with k=1, corresponds to an \mathrm{SU}(2)-bundle embeddable in larger gauge groups, stabilizing solutions to the Yang-Mills equations and influencing Donaldson invariants.^[22]

Structure Group Reduction

Reduction Process

The reduction of the structure group of a principal G-bundle P \to M to a closed subgroup H \subset G is achieved via an H-equivariant bundle map f: P \to Q, where Q \to M is a principal H-bundle, such that the following diagram commutes:

\begin{CD} P @>f>> Q \\ @VVV @VVV \\ M @= M \end{CD}

This map induces an isomorphism of principal G-bundles P \cong Q \times_H G, where the right-hand side is the principal G-bundle obtained by extending the structure group of Q via the inclusion H \hookrightarrow G.^[23] Equivalently, such a reduction exists if and only if the associated bundle P \times_G (G/H) \to M, known as the coset bundle, admits a global section \sigma: M \to P \times_G (G/H). This section corresponds to the H-equivariant map f, as the fiber over each point in M selects a coset in G/H, effectively specifying the reduction locally and globally when consistent. The existence of \sigma can be obstructed by elements in the cohomology groups H^k(M; \mathcal{A}), where \mathcal{A} are the adjoint representations associated to the Lie algebra of G/H, corresponding to the vanishing of certain characteristic classes of P that lie outside the image from BH to BG.^[24]^[25] The process begins by selecting a candidate reduction map, typically via local trivializations of P and adjustment of transition functions to values in H, ensuring H-equivariance under the right actions. Verification involves checking that the map preserves the bundle structure and commutes with the group actions, which follows from the section property. Once reduced, the new principal H-bundle Q allows construction of associated bundles E' = Q \times_H V for H-representations on vector spaces V, which are isomorphic to the original associated bundles E = P \times_G [W](/page/W) for induced G-representations W, via the extension H \to G.^[25]^[23] In modern treatments, particularly for smooth bundles with connections, reduction criteria often involve the holonomy group. The Ambrose–Singer theorem asserts that for a connection \nabla on P, the structure group reduces to the (restricted) holonomy group \mathcal{H}(x) \subset G at each base point x \in M, generated by the curvature form and parallel transport along loops. Specifically, parallel transport along paths defines horizontal lifts, and if the holonomy representation factors through H, the connection pulls back to a connection on the reduced bundle Q, enabling the equivariant map via horizontal subspaces. This provides a differential-geometric condition complementary to topological obstructions, applicable when the holonomy lies in a conjugate of H.^[26]^[27]

Associated Bundles in Reductions

When the structure group of a principal G-bundle P over a base space X is reduced to a closed subgroup H \subset G, the associated bundle E = P \times_\rho V, where \rho: G \to \mathrm{Aut}(V) is a representation on a vector space V, transforms to E' = Q \times_{\rho|_H} V. Here, Q \subset P is the reduced principal H-bundle, and the restriction \rho|_H acts on the same fiber V, potentially trivializing sections or altering the bundle's geometric properties while preserving the total space and base. This reduction is equivalent to the existence of a section in the associated bundle P \times_G (G/H), where G/H is the homogeneous space serving as the fiber, confirming the H-structure on P.^[28] If the original representation \rho factors through H, the associated bundle inherits an induced H-representation directly, simplifying computations in the reduced setting. Alternatively, the reduction produces a new associated bundle P \times_G (G/H), which is a fiber bundle with homogeneous fiber G/H and structure group H, facilitating the study of symmetries preserved under the reduction. For instance, in the orientable reduction from \mathrm{GL}(n, \mathbb{R}) to \mathrm{SO}(n) on the frame bundle of a Riemannian manifold, a compatible metric induces the reduction, yielding associated bundles like the oriented tangent bundle that support Dirac operators via further spin reductions from \mathrm{SO}(n) to \mathrm{Spin}(n), where the spinor bundle P \times_{\mathrm{Spin}(n)} S arises with fiber the spinor representation S.^[28]^[28] In topology, such reductions classify bundles up to isomorphism via homotopy classes of maps from X to classifying spaces, where a reduction to H corresponds to a lift through the map \mathrm{[BH](/page/BH)} \to \mathrm{[BG](/page/BG)} induced by the inclusion H \hookrightarrow G. Stable reductions, relevant for vector bundles in the stable range, are classified by maps to the stable classifying space \mathrm{[BO](/page/BO)} or \mathrm{[BU](/page/BU)}, capturing equivalence classes under Whitney sums and enabling obstructions to further reductions like spin structures through cohomology groups.^[28]^[28]

Properties

Equivalence and Isomorphisms

Two associated bundles E = P \times_\rho V and E' = Q \times_\sigma W over the same base manifold M, where P \to M and Q \to M are principal G-bundles and \rho: G \to \mathrm{Aut}(V), \sigma: G \to \mathrm{Aut}(W) are actions (or representations for vector spaces), are isomorphic as fiber bundles if there exists a fiber-preserving bundle isomorphism \phi: E \to E' covering the identity on M. This isomorphism is compatible with the bundle structures if it intertwines the respective actions, though for general fibers this reduces to preserving the fiber type. In the case of vector bundles (where V, W are vector spaces and \rho, \sigma are linear representations), the isomorphism must be fiberwise linear.^[28] A key criterion for isomorphism arises from the principal bundles: if P and Q are isomorphic as principal G-bundles via a G-equivariant map \psi: P \to Q covering the identity on M, and if the representations \rho and \sigma are equivalent (i.e., there exists an isomorphism \tau: V \to W such that \tau \circ \rho(g) = \sigma(g) \circ \tau for all g \in G), then E \cong E' via the induced map [\psi, \tau]: [p, v] \mapsto [\psi(p), \tau(v)]. This functoriality ensures that isomorphisms of principal bundles lift uniquely to isomorphisms of associated bundles for fixed representations. Conversely, for vector bundles, every associated vector bundle E uniquely determines its frame bundle, the principal \mathrm{GL}(V)-bundle P such that E = P \times_{\mathrm{std}} V, where \mathrm{std} is the standard representation; thus, E is isomorphic to E' if and only if their frame bundles are isomorphic as principal bundles.^[28]^[29] Classification of associated bundles up to isomorphism follows from that of principal bundles. The isomorphism classes of principal G-bundles over M are in bijection with homotopy classes [M, BG], where BG is the classifying space of G. For an associated bundle E = P \times_\rho V, its isomorphism class is thus determined by the class [P] \in [M, BG] together with the equivalence class of the representation \rho (up to conjugacy in \mathrm{Aut}(V)). In the vector bundle case, this specializes to classification by maps to the Grassmannian or B\mathrm{GL}(n), with the associated bundle inheriting the topological invariants of P via \rho.^[28]^[3] For complex vector bundles, classification is often achieved via characteristic classes, particularly Chern classes c_k(E) \in H^{2k}(M; \mathbb{Z}), which are pullbacks of universal Chern classes from B\mathrm{GL}(n, \mathbb{C}) under the classifying map corresponding to the frame bundle. These classes provide obstructions to isomorphism: if two complex vector bundles have different Chern classes, they cannot be isomorphic. However, bundles with matching Chern classes are not necessarily isomorphic, as counterexamples exist over certain bases like spheres. Rationally, the Chern character classifies stable isomorphism classes. Similar obstructions apply using Stiefel-Whitney classes for real bundles. These classes for E are induced from those of the principal bundle via the representation: for instance, the Chern character \mathrm{ch}(E) = \sum \frac{c_k(E)}{k!} transforms under \rho as a polynomial on the Lie algebra invariants.^[28]^[30] In the vector bundle setting, a modern perspective involves stable equivalence: two associated vector bundles E and E' are stably equivalent if E \oplus \epsilon^k \cong E' \oplus \epsilon^l for trivial bundles \epsilon^m of ranks k, l, corresponding to direct sums of representations \rho \oplus \mathrm{id}^k \sim \sigma \oplus \mathrm{id}^l. Stable isomorphism classes are classified by the reduced K-group \tilde{K}(M) (complex) or \tilde{KO}(M) (real), where the class [E] - \mathrm{rk}(E) captures the topology modulo stables; this is particularly useful for oriented bundles where Adams operations refine the classification.^[28]

Canonical Morphisms

In the context of a principal G-bundle P \to M and a right G-space F, the associated bundle is constructed as E = (P \times F)/G \to M, where G acts diagonally via the principal right action on P and the given action on F. The canonical projection \pi_E: E \to M is induced by the quotient map q: P \times F \to E, defined by (p, f) \mapsto [p, f], and the principal projection \pi_P: P \to M; specifically, \pi_E \circ q = \mathrm{pr}_M \circ (\pi_P \times \mathrm{id}_F), where \mathrm{pr}_M is the projection onto the base. This ensures that \pi_E is a fiber bundle with typical fiber F.^[14] A key inclusion arises through the correspondence between sections of E and G-equivariant maps from P to F. Given a G-equivariant smooth map \sigma: P \to F, it induces a section s_\sigma: M \to E via m \mapsto [\pi_P^{-1}(m), \sigma(p)] for any p \in \pi_P^{-1}(m), independent of the choice of p due to equivariance. Conversely, any section s: M \to E lifts to an equivariant map \tilde{s}: P \to F by selecting representatives. This bijection is natural and functorial in the bundle data, particularly evident when F = \mathbb{R}^n and P = \mathrm{Fr}(E) is the frame bundle of a vector bundle E, where equivariant maps correspond to linear frame selections defining sections of E.^[12] For the adjoint representation \mathrm{Ad}: G \to \mathrm{Aut}(\mathfrak{g}), where \mathfrak{g} is the Lie algebra of G, the associated adjoint bundle is \mathrm{Ad}(P) = P \times_G \mathfrak{g} \to M. There is a canonical bundle isomorphism \mathrm{Ad}(P) \cong \mathrm{End}(E) when P = \mathrm{Fr}(E) is the frame bundle of a rank-n vector bundle E \to M with structure group GL(n, \mathbb{R}), and E is the associated bundle via the standard representation. This map sends an element [\beta, A] \in \mathrm{Ad}(P)_m, where \beta \in \mathrm{Fr}(E)_m is a frame and A \in \mathfrak{gl}(n, \mathbb{R}), to the endomorphism L \in \mathrm{End}(E)_m whose matrix with respect to \beta is A; it is well-defined since basis changes transform A via the adjoint action, matching the transition functions of both bundles.^[31]^[32] Forgetful morphisms appear in the context of structure group reduction. Given a reduction of P to a principal H-bundle Q \hookrightarrow P for H \subset [G](/page/G), and a representation \rho: H \to \mathrm{GL}(V), the induced G-representation \mathrm{Ind}_H^G \rho: [G](/page/G) \to \mathrm{GL}(\mathrm{Ind}_H^G V) yields an associated G-bundle E_G = P \times_G \mathrm{Ind}_H^G V \to M, while the H-associated bundle is E_H = Q \times_H V \to M. There is a natural G-equivariant inclusion Q \hookrightarrow P inducing a bundle embedding i: E_H \to E_G over M, viewing E_H as the subbundle corresponding to the H-invariant subspace. Additionally, forgetting the G-action on F yields the trivial bundle morphism E \to M \times F, though this is generally not a fiber bundle map unless E is trivial; it projects orbits to fixed points when possible.^[12] Pullbacks provide another class of canonical morphisms preserving the associated structure. For a smooth map f: N \to M and associated bundle E = P \times_G F \to M, the pullback principal bundle is f^*P = \{(n, p) \in N \times P \mid f(n) = \pi_P(p)\} \to N, with induced G-action, and the associated pullback bundle is f^*E = (f^*P) \times_G F \cong f^*(P \times_G F) \to N. This isomorphism is natural, commuting with the quotient maps, and preserves sections and endomorphisms; for instance, if E is a vector bundle, f^*E inherits the vector structure via the pulled-back representation.^[14]^[12]

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If F is a principal homogeneous H-space on which G acts H-equivariantly, the associated bundle P G F is a principal H-bundle. ... B be a principal G-bundle.
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[PDF] Physics Connections on Discrete Fibre Bundles
The fibres are the orbits of this group, and the base space is the quotient of the bundle by the group action. There is therefore one point in the base space ...
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[PDF] Version 2.2, November 2017 Allen Hatcher Copyright c 2003 by ...
of paint to the point of the Möbius band to which it adheres. Since E ... example, SO(2) is just a circle, since orientation-preserving isometries of ...
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[PDF] Connections on Principal Fibre Bundles
We know reach the idea of an associated bundle. These can be thought of as derived from a principal fibre bundle by means of re- placing the fibre G with ...
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[PDF] The Yang-Mills Equations: A PDE PERSPECTIVE - UC Berkeley math
Given a principal G-bundle P and a vector space V with a G-representation, we may define an associated bundle: E = P ×G V := P × V/(p · g, v) ∼ (p, g · v),.
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[PDF] Meet spin geometry Spin structures and spin manifolds
We can reduce the group further to SO(n) assuming orientability of E. For our oriented vector bundle, there is an associated prin- cipal SO(n)-bundle over B, ...
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[PDF] Dirac operators, Clifford algebras, Spin structures.
Pspin, then the action of Spin(n) on Am can be used to define the associated bundle Pspin spin Am. Thisisthe spinor bundle S. ×. M. Moreover, there is an ...
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Physics Stable Vector Bundles and Instantons* - Project Euclid
In the language of differential geometry, this problem can be phrased as follows: find all possible connections with self-dual curvature on a smooth SU(2)- ...
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[PDF] Self-Duality in Four-Dimensional Riemannian Geometry
Connections can always be viewed in two ways: as defined on principal bundles, or vector bundles. We shall use both methods, so we begin by reviewing the ...
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[PDF] The Topology of Fiber Bundles Lecture Notes Ralph L. Cohen Dept ...
A fiber bundle is a parameterized family of objects, each isomorphic to a fiber, where the family is parameterized by points in the base space.
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reduction and lift of structure groups in nLab
Jul 18, 2024 · For G → K G \to K a monomorphism of groups, a G G -structure on a K K -principal bundle is a reduction of the structure group from K K to G G .
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[PDF] Reduction of structure group
Suppose P is a principal G-bundle, and H ⊂ G is a subgroup (not necessarily normal). Then we have a “short exact sequence”. 0 → H → G → G/H → 0. (When H ...
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[PDF] connections
The Ambrose-. Singer theorem states that the structure group of any principal G-bundle P with a connection form can be reduced to the holonomy group at each ...
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[PDF] Kobayashi-Nomizu.pdf
By Proposition 5.6, we see that the structure group of L(M) can be reduced to the orthogonal group O(n), provided that M is paracompact. Example 5.6. More ...
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[PDF] Fibre Bundles
20 HUSEMOLLER. Fibre Bundles. 3rd ed. 21 HUMPHREYS. Linear Algebraic Groups ... principal bundle and Chapter 19 on the definition of Chern classes by ...
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[PDF] bundles, classifying spaces and characteristic classes
the terminology may be somewhat.<|separator|>
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Associated vector bundles and Characteristic Classes - MathOverflow
Jun 29, 2016 · The Chern classes of the associated bundles E and F are just the pullbacks of the associated bundles to ρ and ρ′ on BG. So, the Chern ...Characteristic classes of lifted bundles - MathOverflowCharacteristic classes for $E_8$ bundles - MathOverflowMore results from mathoverflow.netMissing: Shiing- Shen
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[PDF] arXiv:2312.05711v2 [math.AG] 12 Dec 2024
Dec 12, 2024 · Under this identification, the adjoint bundle of a principal GLn-bundle is canonically isomorphic to the endomorphism bundle of the ...
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[PDF] Chern-Weil Constructions on ΨDO Bundles - arXiv
GL(n, C), then the frame bundle P := GL(E) → B is a principal GL(n, C)-bundle and. Ad P is isomorphic to End(E), the endomorphism bundle of E. If E is an ...