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Principal bundle

A principal bundle, also known as a principal G-bundle for a Lie group G, is a fiber bundle P \to M in which the structure group G acts freely and transitively on the total space P from the right, with the projection \pi: P \to M being G-invariant, meaning \pi(p \cdot g) = \pi(p) for all p \in P and g \in G, and the bundle is locally trivial via G-equivariant diffeomorphisms to open sets in M times G.^[1]^[2] This structure ensures that the fibers over each point in the base manifold M are diffeomorphic to G itself, and the quotient space P/G is homeomorphic (or diffeomorphic, in the smooth case) to M.^[1] Principal bundles generalize Cartesian products of a base space with a group while capturing twisting or non-trivial topology, and they form the foundation for associated vector bundles, where representations of G on vector spaces yield fibers modeled on those spaces.^[2] A key property is that a principal G-bundle admits a global section if and only if it is trivial, i.e., isomorphic to the product bundle M × G.^[1]^[2] They are locally trivial, covered by open sets U_α ⊂ M with equivariant trivializations φ_α: π⁻¹(U_α) → U_α × G satisfying φ_α(p · g) = (π(p), g' · g) for some adjustment by transition functions in G.^[1] Classic examples include the frame bundle of a smooth manifold M, which is a principal GL(n, ℝ)-bundle whose sections correspond to bases of the tangent spaces at each point, and its reduction to an SO(n)-bundle for Riemannian manifolds with a metric.^[1] Another is the tautological S¹-bundle over complex projective space ℂℙⁿ, arising from the action of U(1) on unit spheres.^[1] In differential geometry, principal bundles are essential for defining connections, which are G-equivariant ℝⁿ-valued 1-forms on P that split the tangent bundle into horizontal and vertical subbundles, enabling the study of curvature and parallel transport.^[3] Beyond pure mathematics, principal bundles provide the geometric framework for gauge theories in physics, where the total space P models the configuration space of gauge fields, and connections represent gauge potentials, as in Yang-Mills theory with structure groups like SU(3) for quantum chromodynamics.^[3] Their classification up to isomorphism is governed by homotopy classes in the classifying space BG, linking them to characteristic classes and topological invariants.^[2]

Fundamentals

Formal definition

A principal G-bundle, where G is a Lie group, is a fiber bundle (P, \pi, M) with fiber G over a smooth manifold M, consisting of a smooth manifold P, a surjective submersion \pi: P \to M, and a smooth right action of G on P that is free and transitive on each fiber \pi^{-1}(m).^[4] The action is denoted by p \cdot g for p \in P and g \in G, and it is compatible with the projection in the sense that \pi(p \cdot g) = \pi(p) for all p \in P and g \in G.^[4] This structure ensures that each fiber \pi^{-1}(m) is a G-torsor, meaning it admits a unique free and transitive right G-action up to isomorphism.^[5] The bundle is locally trivial: for every point m \in M, there exists an open neighborhood U \subset M and a G-equivariant diffeomorphism \phi: \pi^{-1}(U) \to U \times G satisfying \phi(p \cdot g) = (\pi(p), \phi(p)_2 \cdot g), where \phi(p) = (\pi(p), \phi(p)_2) and the action on U \times G is defined by (u, h) \cdot g = (u, h g).^[4] In contrast to a general fiber bundle, where the fibers are simply diffeomorphic to a fixed model space F with an effective action of a structure group on F, a principal bundle specifies F = G and requires the action to be free and transitive, thereby canonically identifying the fibers with the group itself via the group action.^[6]

Examples

The trivial principal bundle provides the simplest example of a principal bundle structure. Given a manifold M and a Lie group G, the product space P = M \times G forms a principal G-bundle over M via the projection \pi: P \to M defined by \pi(m, g) = m, with the right G-action given by (m, g) \cdot h = (m, gh) for h \in G. This construction is locally trivial everywhere, as the bundle is globally a product, and it serves as the model for understanding local behavior in more general principal bundles. A fundamental example in differential geometry is the frame bundle of a vector bundle. For a real vector bundle E \to M of rank n, the frame bundle P(E) \to M is the principal \mathrm{GL}(n, \mathbb{R})-bundle whose fiber over each point m \in M consists of all ordered bases (frames) of the fiber E_m.^[7] The right action of \mathrm{GL}(n, \mathbb{R}) on P(E) changes bases via matrix multiplication, and this bundle captures the linear structure of E while being locally trivial over coordinate charts where bases can be chosen consistently.^[7] The Hopf fibration exemplifies a non-trivial principal bundle with compact fibers. The classical Hopf fibration is the principal U(1)-bundle S^3 \to S^2 with fiber S^1, where S^3 is the total space and the projection identifies points differing by phase multiplication in \mathbb{C}^2.^[8] More generally, the fibration S^{2k+1} \to \mathbb{CP}^k for k \geq 1 is a principal U(1)-bundle, with fibers consisting of scalar multiples in \mathbb{C}^{k+1}, illustrating how complex projective spaces arise as base spaces for circle bundles.^[8] In physics, principal bundles model gauge symmetries underlying fundamental interactions. For electromagnetism, the gauge fields are connections on a principal U(1)-bundle over spacetime, where sections correspond to phase choices for charged fields.^[9] Similarly, Yang-Mills theories employ principal SU(n)-bundles over spacetime for non-abelian gauge groups, such as SU(3) in quantum chromodynamics, where the bundle structure encodes the freedom in choosing local gauge transformations.^[9] Stiefel manifolds appear as total spaces of orthogonal frame bundles over Grassmannians. The real Stiefel manifold V_{k,n}(\mathbb{R}), consisting of orthonormal k-frames in \mathbb{R}^n, is the total space of the principal O(k)-bundle over the Grassmannian \mathrm{Gr}_{k,n}(\mathbb{R}) of k-planes in \mathbb{R}^n, with the projection mapping each frame to its span and the right O(k)-action rotating the frame within the plane.^[10] This construction highlights the role of principal bundles in parametrizing oriented subspaces.^[2]

Structural Properties

Trivializations and sections

A trivialization of a principal G-bundle (P, \pi, M) over an open subset U \subseteq M is a G-equivariant diffeomorphism \phi: \pi^{-1}(U) \to U \times G satisfying \mathrm{pr}_1 \circ \phi = \pi|_{\pi^{-1}(U)}, where \mathrm{pr}_1 is the projection onto the first factor and G acts on U \times G by right multiplication (u, h) \cdot k = (u, h k).^[1] The equivariance condition requires \phi(p \cdot g) = \phi(p) \cdot g for all p \in \pi^{-1}(U) and g \in G.^[1] By definition, every principal bundle admits a cover \{U_\alpha\} of M by open sets each equipped with such a trivialization \phi_\alpha.^[2] Given two trivializations \phi_i and \phi_j over overlapping open sets U_i and U_j, the transition function g_{ij}: U_i \cap U_j \to G is defined by \phi_j \circ \phi_i^{-1}(u, h) = (u, g_{ij}(u) h) for u \in U_i \cap U_j and h \in G. These transition functions satisfy the cocycle condition g_{ij}(u) g_{jk}(u) = g_{ik}(u) on triple overlaps U_i \cap U_j \cap U_k, ensuring consistency across the cover.^[1] Trivializations over the same open set are unique up to right G-action, meaning if \phi and \psi are two trivializations of \pi^{-1}(U), then there exists a map f: U \to G such that \psi(p) = \phi(p) \cdot f(\pi(p)) for all p \in \pi^{-1}(U).^[2] A (global) section of the principal bundle is a smooth map s: M \to P such that \pi \circ s = \mathrm{id}_M.^[5] The existence of a global section implies that the bundle is trivial, as it induces an isomorphism M \times G \to P via (m, g) \mapsto s(m) \cdot g.^[5] Conversely, every trivial principal bundle admits global sections.^[5] Since the right G-action on P is free, for any section s we have s(m) \cdot g = s(m) only if g = e, the identity element.^[5] Local sections arise naturally from trivializations, for instance, the constant section over U_\alpha given by \phi_\alpha^{-1}(u, e) for u \in U_\alpha.^[1]

Characterization of smooth principal bundles

In differential geometry, a smooth principal bundle over a smooth manifold M consists of a total space P, which is also a smooth manifold, together with a smooth submersion \pi: P \to M and a Lie group G acting smoothly, freely, and properly on P from the right such that the action is fiber-preserving (i.e., \pi(p \cdot g) = \pi(p) for all p \in P and g \in G).^[11]^[8] The fibers \pi^{-1}(m) for m \in M are thus diffeomorphic to G via the transitive action, and the smoothness of the submersion ensures that local sections exist, making the fibers embedded smooth submanifolds.^[12] This structure distinguishes smooth principal bundles by imposing compatibility with the differential topology of the underlying manifolds. Two smooth principal bundles (P, \pi, M, G) and (P', \pi', M', G) are equivalent if there exists a smooth G-equivariant diffeomorphism f: P \to P' such that \pi' \circ f = \phi \circ \pi for some diffeomorphism \phi: M \to M'.^[13]^[11] Equivalence in this sense preserves the smooth structure and the right G-action, which is right-invariant by definition: for all p \in P and g, h \in G, (p \cdot g) \cdot h = p \cdot (gh), with the multiplication in G smooth.^[8] This invariance ensures that the action respects the manifold structures and allows the bundle to be reconstructed from its local trivializations. A smooth principal bundle admits an atlas of trivializations \{\phi_i: \pi^{-1}(U_i) \to U_i \times [G](/page/G)\}, where \{U_i\} is an open cover of M and each \phi_i is a [G](/page/G)-equivariant diffeomorphism, with the right action on U_i \times [G](/page/G) given by (u, k) \cdot g = (u, k g).^[12]^[11] The transition functions g_{ij}: U_i \cap U_j \to [G](/page/G), defined by \phi_j = (\phi_i \times \mathrm{id}_G) \circ ( \mathrm{id}_{U_i \cap U_j} \times g_{ij} ), are smooth maps satisfying the cocycle condition g_{ij}(u) g_{jk}(u) = g_{ik}(u) for u \in U_i \cap U_j \cap U_k.^[8] These smooth transition functions encode the twisting of the bundle and ensure global consistency in the smooth category. In contrast to topological principal bundles, where the total space, base, projection, and group action are merely continuous and the transition functions are continuous maps to the topological group G, the smooth version requires all components—manifolds P and M, submersion \pi, Lie group G, action, and transition functions—to be smooth.^[11] This added smoothness ensures compatibility with the tangent spaces and differential forms on P and M, enabling the construction of smooth connections and other tools central to differential geometry.^[8]

Applications

Reduction of the structure group

In the context of a principal G-bundle P \to M over a smooth manifold M, a reduction of the structure group to a closed subgroup H \subset G is given by a principal H-subbundle Q \subset P such that P = Q \cdot G, where Q \cdot G denotes the saturation of Q under the right G-action on P, ensuring that every G-orbit in P intersects Q.^[2] This construction simplifies the geometry of P by restricting the fiberwise action to the smaller group H, while preserving the bundle structure over M.^[2] Equivalently, such a reduction exists if and only if the classifying map f: M \to BG for P lifts (up to homotopy) to a map g: M \to BH composing with the induced BH \to BG to yield f, assuming H is a closed subgroup of the Lie group G.^[14] In terms of cocycles, if \{g_{ij}\} are the G-valued transition functions of P on an open cover \{U_i\} of M, the reduction corresponds to the existence of an H-valued cocycle \{h_{ij}\} refining \{g_{ij}\}, meaning there exist G-valued functions u_i on U_i such that g_{ij} = u_i h_{ij} u_j^{-1} for all i,j.^[15] More generally, when considering the normalizer N_G(H) = \{k \in G \mid k H k^{-1} = H\}, the refinement takes the form g_{ij} = k_{ij} h_{ij} l_{ij} with k_{ij}, l_{ij} \in N_G(H), allowing for conjugate adjustments within the normalizer.^[16] A maximal reduction of the structure group occurs when H is as large as possible while admitting such a subbundle; common cases include reduction to the identity component G_0 of G (preserving connectedness) or to the center Z(G) (capturing central extensions).^[2] For instance, the frame bundle P(M, \mathrm{GL}(n,\mathbb{R})) of an n-dimensional paracompact manifold M admits a reduction to \mathrm{O}(n) if and only if M carries a Riemannian metric, which is always possible and corresponds to an \mathrm{O}(n)-structure defining the metric via the associated orthogonal bundle.^[2] Reductions to H are in bijective correspondence with G/H-bundle structures over M: specifically, the associated bundle P \times_G (G/H) \to M admits a global section if and only if P reduces to H, with the section identifying the G/H-fibers pointwise via the H-orbits in G.^[14] This equivalence underscores the role of reductions in classifying bundle geometries through homogeneous spaces G/H.^[2]

Associated bundles and frames

Given a principal G-bundle \pi: P \to M and a representation \rho: G \to \mathrm{GL}(V) on a vector space V, the associated vector bundle E = P \times_\rho V is constructed as the quotient space (P \times V)/G, where the G-action is defined by (p, v) \cdot g = (p \cdot g, \rho(g^{-1}) v) for p \in P, v \in V, and g \in G.^[17] The equivalence relation identifies [p, v] \sim [p \cdot g, \rho(g^{-1}) v], and the projection map is \pi_E: E \to M given by \pi_E([p, v]) = \pi(p), ensuring E is a vector bundle over M with fibers isomorphic to V.^[17] The tangent bundle TM of an n-dimensional smooth manifold M exemplifies this association: the frame bundle FM is the principal \mathrm{GL}(n, \mathbb{R})-bundle over M whose fiber at m \in M consists of all ordered bases (frames) of T_m M, with right action f \cdot g = (g_{i1} f_1, \dots, g_{in} f_n) for f = (f_1, \dots, f_n) and g = (g_{ij}) \in \mathrm{GL}(n, \mathbb{R}).^[18] Then, TM is the associated vector bundle FM \times_{\mathrm{GL}(n, \mathbb{R})} \mathbb{R}^n, where [f, v] \mapsto \sum_i f_i v^i \in T_{\pi(f)} M for v = (v^i) \in \mathbb{R}^n.^[18] If M admits a Riemannian metric, reduction of the structure group of FM to the orthogonal group \mathrm{O}(n) yields a principal \mathrm{O}(n)-bundle whose associated bundle consists of orthonormal frames.^[18] Global sections of the associated bundle E correspond bijectively to \rho-equivariant maps \sigma: P \to V, i.e., maps satisfying \sigma(p \cdot g) = \rho(g^{-1}) \sigma(p) for all p \in P, g \in G.^[17] Locally, over a trivialization U \subset M with section s: U \to \pi^{-1}(U), a section of E|_U is represented by [s(m), v(m)] for a smooth v: U \to V, and the equivariance ensures consistency under gauge transformations.^[17] Thus, \Gamma(E) \cong \{ \tilde{s}: P \to V \mid \tilde{s}(p \cdot g) = \rho(g^{-1}) \tilde{s}(p) \ \forall g \in G \}.^[17] Associated bundles inherit functoriality from principal bundles: for a smooth map f: N \to M, the pullback principal bundle f^* P = N \times_M P induces the pullback associated bundle f^* E = (f^* P) \times_\rho V, with fibers over n \in N given by \{ (n, [p, v]) \mid f(n) = \pi(p) \}, preserving the vector bundle structure.^[2] Conversely, for a surjective submersion f: N \to M and associated bundle E over N, the pushforward f_* E over M has fiber over m \in M consisting of smooth sections of E over f^{-1}(m), equipped with a natural vector bundle structure when f is proper.^[2]

Connections on principal bundles

A connection on a principal bundle provides a geometric framework for defining notions of parallel transport and differentiation, enabling the study of how objects transform along paths in the base manifold. In the context of a smooth principal G-bundle \pi: P \to M with structure group G, an Ehresmann connection is defined as a smooth horizontal distribution H \subset TP such that for each p \in P, H_p is a subspace of T_p P complementary to the vertical subspace V_p = \ker(d\pi_p), and the projection d\pi|_{H_p}: H_p \to T_{\pi(p)} M is a linear isomorphism, with the distribution being right-invariant under the G-action: R_g^* H_p = H_{p \cdot g} for all g \in G.^[19] This setup ensures a consistent choice of "horizontal" directions transverse to the fibers, facilitating the lifting of curves from M to P.^[20] The vertical subspace V_p at p \in P is the kernel of the differential d\pi_p: T_p P \to T_{\pi(p)} M, which coincides with the tangent space to the fiber \pi^{-1}(\pi(p)) and is spanned by the fundamental vector fields \{ p \cdot \xi \mid \xi \in \mathfrak{g} \}, where \mathfrak{g} is the Lie algebra of G and p \cdot \xi denotes the infinitesimal action of \xi at p.^[19] Thus, V_p \cong \mathfrak{g} as vector spaces, providing a canonical identification that underpins the bundle's G-structure. The decomposition T_p P = V_p \oplus H_p then splits the tangent bundle into vertical and horizontal components, with the horizontal part encoding the connection's geometric data.^[21] Equivalently, an Ehresmann connection can be described by a \mathfrak{g}-valued 1-form \omega \in \Omega^1(P, \mathfrak{g}), called the connection form, satisfying two key properties: it reproduces the Lie algebra elements on vertical vectors, \omega(\xi^\#_p) = \xi for \xi \in \mathfrak{g} where \xi^\#_p = p \cdot \xi, and it is equivariant under the right G-action, R_g^* \omega = \mathrm{Ad}(g^{-1}) \omega for g \in G.^[19] The horizontal subspace is then the kernel H_p = \ker \omega_p, ensuring that \omega projects tangent vectors onto their vertical components relative to \mathfrak{g}. This formulation captures the connection's tensorial nature and allows for local expressions in trivializations of P. The curvature of the connection measures the extent to which the horizontal distribution fails to be integrable and is given by the \mathfrak{g}-valued 2-form \Omega = d\omega + \frac{1}{2} [\omega, \omega], where [\cdot, \cdot] denotes the Lie bracket in \mathfrak{g} extended to forms via the wedge product.^[19] More precisely, \Omega is the horizontal part of d\omega, satisfying \Omega(X, Y) = d\omega(X, Y) + \frac{1}{2} [\omega(X), \omega(Y)] for horizontal vectors X, Y, and it transforms as R_g^* \Omega = \mathrm{Ad}(g^{-1}) \Omega, making it a tensorial form of type \mathrm{Ad} G.^[22] A connection is flat if \Omega = 0, in which case the horizontal distribution defines an integrable foliation, but in general, \Omega quantifies holonomy obstructions.^[19] Parallel transport along a curve c: [0,1] \to M is defined by lifting c horizontally in P: for p \in \pi^{-1}(c(0)), there exists a unique horizontal curve \tilde{c}: [0,1] \to P with \tilde{c}(0) = p and \pi \circ \tilde{c} = c, provided the connection is smooth.^[20] This induces a G-equivariant isomorphism \tau_c: \pi^{-1}(c(0)) \to \pi^{-1}(c(1)) between fibers, preserving the bundle structure and defining the holonomy of the connection, which encodes global geometric information via the image of the holonomy group.^[5]

Classification

Topological classification

The topological classification of principal bundles concerns the determination of isomorphism classes of such bundles over a base space M, where isomorphism is understood in the category of topological spaces without additional structure. For a topological group G, the set of isomorphism classes of principal G-bundles over M is in bijective correspondence with the homotopy classes of continuous maps [M, BG], where BG is the classifying space of G. This classifying space BG is characterized as the base space of the universal principal G-bundle EG \to BG, which is contractible as a total space and thus serves as a universal model for all principal G-bundles via pullback. The bijection arises from the fact that any principal G-bundle P \to M admits a classifying map f: M \to BG such that P \cong f^* EG, with two bundles isomorphic if and only if their classifying maps are homotopic.^[23] When G is a discrete group, the classifying space BG is a K(G, 1)-space, meaning its higher homotopy groups vanish, and the homotopy classes [M, BG] coincide with the first Čech cohomology group \check{H}^1(M; \underline{G}), where \underline{G} denotes the constant sheaf associated to G. In this case, principal G-bundles over M are classified by Čech 1-cocycles with values in G, representing transition functions on an open cover of M, up to coboundaries. More generally, for arbitrary topological G, the classification via [M, BG] can be refined using sheaf cohomology in the category of sheaves of sets or groups on M, where the isomorphism classes correspond to elements in the first cohomology group of the sheaf of G-principal bundles. This cohomological perspective unifies the topological data encoded in the bundle's transition functions with global homotopy invariants.^[24] A concrete method to construct and classify principal bundles over spheres or manifolds decomposable as unions of cells is the clutching construction. For a base space M = U \cup V where U and V are open sets homeomorphic to disks or balls, a principal G-bundle over M is obtained by taking trivial bundles over U and V and gluing them along the intersection via a clutching map \phi: U \cap V \simeq S^{n-1} \to G. Two such bundles are isomorphic if their clutching maps are homotopic in G. On spheres S^n, this reduces the classification to homotopy classes [\pi_{n-1}(G)], with the clutching map providing an explicit representative of the bundle's class in [S^n, BG] \simeq \pi_{n-1}(G). This construction highlights how local trivializations determine global topology through boundary data.^[25]^[26] For the specific case of S^1-principal bundles, which are circle bundles, the classification is given by the first Chern class c_1 \in H^2(M; \mathbb{Z}), an element of the second integer cohomology group of M. The Chern class arises as the primary characteristic class associated to the bundle via the classifying map to BS^1 \cong \mathbb{C}P^\infty, and it detects the bundle's twisting: trivial bundles correspond to c_1 = 0, while non-trivial examples include the Hopf fibration over S^2 with c_1 generating H^2(S^2; \mathbb{Z}). Isomorphic bundles share the same Chern class, providing a complete invariant for oriented circle bundles over paracompact bases.^[23] In general, the existence of a principal G-bundle or its isomorphism class can be probed using obstruction theory, which measures the failure of a partial classifying map or section to extend globally. Assuming a partial map defined on the k-skeleton of a CW-complex model for M, the primary obstruction to extension lies in H^{k+1}(M; \pi_k(BG)), but for bundles, it relates directly to the homotopy groups of G via the long exact sequence of the fibration EG \to BG. Specifically, the primary obstruction to triviality (or existence of a global section) is in H^2(M; \pi_1(G)), with secondary obstructions in H^3(M; \pi_2(G)) if the primary vanishes, and higher terms following the Postnikov tower of BG. This cohomological ladder provides a systematic way to compute when a bundle exists or is unique up to isomorphism based on the topology of M and G.^[23]

Smooth and holomorphic classification

In the smooth category, principal G-bundles over a smooth manifold M, where G is a Lie group, are classified up to smooth isomorphism by the first non-abelian Čech cohomology group \check{H}^1(M, G) with coefficients in smooth G-valued functions on intersections of an open cover. Specifically, a smooth cocycle consists of smooth transition functions g_{ij}: U_i \cap U_j \to G satisfying g_{ij}(x) g_{jk}(x) = g_{ik}(x) for x \in U_i \cap U_j \cap U_k, and two cocycles are equivalent if they differ by a smooth coboundary h_i: U_i \to G via g'_{ij} = h_i^{-1} g_{ij} h_j.^[2] This Čech description captures the local trivializations and ensures the bundle's smooth structure. For G = U(1), the classification aligns with the topological case via the first Chern class in H^2(M; \mathbb{Z}). An equivariant refinement of this classification uses smooth classifying spaces: smooth principal G-bundles over M correspond bijectively to smooth homotopy classes of maps [M, BG]_{\text{smooth}}, where BG is equipped with a smooth structure (e.g., via diffeological spaces for general Lie groups), and the universal bundle EG \to BG pulls back along such maps.^[27] This extends the topological classification by requiring the classifying map to be smooth, ensuring diffeomorphism equivalence of bundles corresponds to smooth homotopies, and applies particularly well when G admits a smooth model for its classifying space. For compact Lie groups, the smooth and topological classifications coincide up to isomorphism due to the existence of smooth partitions of unity on paracompact bases.^[2] Holomorphic principal G-bundles, where G is a complex Lie group, are defined over a complex manifold M using an open cover with holomorphic transition functions g_{ij}: U_i \cap U_j \to G satisfying the cocycle condition holomorphically. These bundles are classified up to holomorphic isomorphism by the non-abelian holomorphic Čech cohomology \check{H}^1(M, G) or, equivalently, by holomorphic maps to the classifying space BG in the holomorphic category. For G = \mathrm{GL}(n, \mathbb{C}), such principal bundles are in bijective correspondence with holomorphic vector bundles of rank n over M, classified by holomorphic cohomology. Characteristic classes provide topological invariants refining the smooth and holomorphic classifications. For principal U(n)-bundles, the Chern classes c_k \in H^{2k}(M, \mathbb{Z}) in even degrees classify the bundles up to isomorphism when the base is a complex manifold, with the total Chern class c(E) = 1 + c_1(E) + \cdots + c_n(E) determined by the curvature of invariant connections via Chern-Weil theory.^[2] Similarly, for principal O(n)-bundles, the Pontryagin classes p_k \in H^{4k}(M, \mathbb{Z}) serve as primary invariants, related to Chern classes of the complexification by p_k = (-1)^k c_{2k}, and together with Stiefel-Whitney classes, they fully classify oriented real bundles in low dimensions. These classes remain invariant under smooth or holomorphic isomorphisms and extend the topological classification by incorporating differential-geometric data.

References

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Oct 28, 2019 · A principal G-bundle is a triple (P,B,π) where π: P → B is a map with a continuous, free right action P × G → P, where π is invariant.
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### Definition and Characterization of Smooth Principal Bundles
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[PDF] arXiv:0707.4136v1 [hep-th] 27 Jul 2007
Jul 27, 2007 · gΩω = Ad(g−1)Ωω. The connection ω is said to be flat if its curvature is 0, and in this case holonomies around null-homotopic loops ...
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Fibre Bundles | SpringerLink
Basic properties, homotopy classification, and characteristic classes of fibre bundles ... The Gauge Group of a Principal Bundle. Dale Husemoller. Pages 79 ...
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[PDF] G-bundles,˘Cech Cohomology and the Fundamental Group
Jan 22, 2015 · In this thesis, we will study G-bundles from an algebraic-topological viewpoint, elucidating a connection between G-bundles over a fixed well- ...
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[PDF] Version 2.2, November 2017 Allen Hatcher Copyright c 2003 by ...
by gluing together two vector bundles over X×D2 by means of a generalized clutching function. We begin by describing this more general clutching construction.Missing: principal | Show results with:principal
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[PDF] The Topology of Fiber Bundles Lecture Notes Ralph L. Cohen Dept ...
We discuss many examples, including covering spaces, vector bundles, and principal bundles. ... . Consider the principal G bundle E over ΣG defined to be trivial ...
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[1709.10517] Smooth classifying spaces - arXiv
Sep 29, 2017 · We develop the theory of smooth principal bundles for a smooth group G, using the framework of diffeological spaces.
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STABILITY OF COMPLEX VECTOR BUNDLES - Project Euclid
Mar 2, 1996 · Introduction. The notion of stability plays a central role in complex and algebraic geometry. It was introduced by D. Mumford [5] and F.