nLab
fiber bundle

Context

Bundles

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

A fibre bundle or fiber bundle is a bundle in which every fibre is isomorphic, in some coherent way, to a standard fibre (sometimes also called typical fiber). Though it is pre-dated by many examples and methods, systematic usage of locally trivial fibre bundles with structure groups in mainstream mathematics started with a famous book of Steenrod.

One may say that ‘fibre bundles are fibrations’ by the Milnor slide trick.

Definitions

In the most general sense, a bundle over an object BB in a category CC is a morphism p:EBp: E \to B in CC.

In an appropriate contexts, a fibre bundle over BB with standard fibre FF may be defined as a bundle over BB such that, given any global element x:1Bx: 1 \to B, the pullback of EE along xx is isomorphic to FF. Certainly this definition is appropriate whenever CC has a terminal object 11 which is a generator, as in a well-pointed category; even then, however, one often wants the more restrictive notion below.

One often writes a typical fibre bundle in shorthand as FEBF \to E \to B or

F E B \array { F & \rightarrow & E \\ & & \downarrow \\ & & B }

even though there is not a single morphism FEF \to E but instead one for each global element xx (and none at all if BB has no global elements!).

If CC is a site, then a locally trivial fibre bundle over BB with standard fibre FF is a bundle over BB with a cover (j α:U αB) α(j_\alpha: U_\alpha \to B)_\alpha such that, for each index α\alpha, the pullback E αE_\alpha of EE along j αj_\alpha is isomorphic in the slice category C/U αC/{U_\alpha} to the trivial bundle U α×FU_\alpha \times F.

One can also drop FF and define a slightly more general notion of locally trivial bundle over BB as a bundle over BB with a cover (j α:U αB) α(j_\alpha: U_\alpha \to B)_\alpha such that, for each index α\alpha, there is a fibre F αF_\alpha and an isomorphism in C/U αC/{U_\alpha} between the pullback E αE_\alpha and the trivial bundle U α×F αU_\alpha \times F_\alpha. Every locally trivial fibre bundle is obviously a locally trivial bundle; the converse holds if BB is connected.

Now suppose that EE is a fibre bundle over BB with standard fibre FF, locally trivialised using the cover (U α) α(U_\alpha)_\alpha. Given an index α\alpha and an index β\beta, let U α,βU_{\alpha,\beta} be the fibred product (pullback) of U αU_\alpha and U βU_\beta. Then we have an automorphism g α,βg_{\alpha,\beta} of U α,β×FU_{\alpha,\beta} \times F in C/U α,βC/{U_{\alpha,\beta}} as follows: (diagram to come) The g α,βg_{\alpha,\beta} are the transition morphisms of the locally trivial fibre bundle EE.

Often one considers special kinds of bundles, by requiring structure on the standard fibre FF and/or conditions on the transition morphisms g α,βg_{\alpha,\beta}. For example:

Special cases

GG-bundle

If GG is a group object in CC that acts on FF, then a GG-bundle (or bundle with structure group GG) over BB with standard fibre FF is a locally trivial fibre bundle over BB with standard fibre FF together with morphisms U α,βGU_{\alpha,\beta} \to G that, relative to the action of GG on FF, give the transition maps g α,βg_{\alpha,\beta}. (The morphism U α,βGU_{\alpha,\beta} \to G is also written g α,βg_{\alpha,\beta}, conflating action with application.)

GG-principal bundles

More specifically, a (right or left) principal GG-bundle over BB is a GG-bundle over BB with standard fibre GG, associated with the action of GG on itself by (right or left) multiplication.

Structure-preserving bundles

If FF is an object of a concrete category over CC, then we can consider locally trivial fibre bundles with standard fibre FF such that the transition morphisms are structure-preserving morphisms. If the automorphism group Aut(F)Aut(F) can be internalised in CC, then this the same as an Aut(F)Aut(F)-bundle, but the concept makes sense in any case.

Vector bundles

As a fairly specific example, if FF is a topological vector space (and CC is a category with structure to support this, such as Top or Diff), then a vector bundle over BB with standard fibre FF is a GL(F)GL(F)-bundle over BB with standard fibre FF, where GL(F)GL(F) is the general linear group with its defining action on FF.

Associated bundles

Given a right principal GG-bundle π:PX\pi: P\to X and a left GG-space FF, all in a sufficiently strong category CC (such as Top), one can form the quotient object P× GF=(P×F)/P\times_G F = (P\times F)/{\sim}, where P×FP \times F is a product and \sim is the smallest congruence such that (using generalized elements) (pg,f)(p,gf)(p g,f)\sim (p,g f); there is a canonical projection P× GFXP\times_G F\to X where the class of (p,f)(p,f) is mapped to π(p)X\pi(p)\in X, hence making P× GFXP\times_G F\to X into a fibre bundle with typical fiber FF, and the transition functions belonging to the action of GG on FF. We say that P× GFXP\times_G F\to X is the associated bundle to PXP\to X with fiber FF.

Generalizations

In higher category theory

In higher category theory the notion of fiber bundle generalizes. See

In commutative algebra

Under the interpretation of modules as generalized vector bundles, locally trivial fiber bundles correspond to locally free modules. See there for more.

In noncommutative geometry

In noncommutative geometry both principal and associated bundles have analogues. The principal bundles over noncommutative spaces typically have structure group replaced by a Hopf algebra; the most well-known class whose base is described by a single algebra are Hopf–Galois extensions; the global sections of the associated bundle are formed using cotensor product. Transition functions can be to some extent emulated using noncommutative localizations, which yield nonaffine generalizations of Hopf–Galois extensions. Another generalization is when Hopf–Galois extensions in the sense of comodule algebras are replaced by entwining structures with analogous Galois condition.

Literature

  • Norman Steenrod, The topology of fibre bundles, Princeton Mathematical Series 14, 1951. viii+224 pp. MR39258; reprinted 1994
  • Dale Husemöller, Fibre bundles, McGraw-Hill 1966 (300 p.); Springer Graduate Texts in Math. 20, 2nd ed. 1975 (327 p.), 3rd. ed. 1994 (353 p.)
  • M M Postnikov, Лекции по геометрии. Семестр IV. Дифференциальная геометрия (Lectures in geometry. Semester IV. Differential geometry.) Nauka, Moscow, 1988. 496 pp. MR90h:53002

Revised on September 16, 2013 00:53:17 by Urs Schreiber (89.204.139.122)