fibergrupper

Fiber groups, in mathematics, refer to the structure group of a fiber bundle. A fiber bundle consists of a total space E, a base space B, a typical fiber F, and a projection π: E → B, together with local trivializations that identify π^{-1}(U) with U × F for open sets U ⊂ B. The structure group G is a topological (often Lie) group that acts on F by homeomorphisms or linear maps, and the bundle is glued together by transition functions g_ij: U_i ∩ U_j → G.

A principal G-bundle is a fundamental example: the fiber is G itself and G acts on itself by left multiplication. Any associated bundle can be formed from a principal G-bundle by choosing a G-space F and taking the quotient E ×_G F, which yields a new fiber bundle with fiber F. The choice of G and its action determines the geometric and topological properties of the bundle.

Common structure groups include GL(n, R), GL(n, C), SO(n), and U(n). The tangent bundle of a smooth manifold has structure group GL(n, R), and the frame bundle carries the canonical GL(n, R) structure group. The Möbius band can be described as a line bundle with a reduced structure group {±1} ⊂ GL(1, R), illustrating how reductions of the structure group reflect additional geometric structure.

Classification and applications: For suitable base spaces X, isomorphism classes of principal G-bundles correspond to homotopy classes [X, BG], where BG is the classifying space of G. This framework underpins gauge theory in physics, where connections on principal bundles model gauge fields; in geometry, structure groups control curvature and holonomy, and reductions encode structures such as orientations, complex structures, or Riemannian metrics.