mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the fiber bundle construction theorem is a

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...

which constructs a

fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...

from a given base space, fiber and a suitable set of

transition functions In mathematics, a transition function may refer to: * a transition map between two charts of an atlas of a manifold or other topological space * the function that defines the transitions of a state transition system in computing, which may refer mor ...

. The theorem also gives conditions under which two such bundles are

isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...

. The theorem is important in the

associated bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces wit ...

construction where one starts with a given bundle and surgically replaces the fiber with a new space while keeping all other data the same.

Formal statement

Let ''X'' and ''F'' be

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

s and let ''G'' be a

topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...

with a continuous left action on ''F''. Given an

open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\s ...

of ''X'' and a set of

continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...

s :

t_ : U_i \cap U_j \to G

defined on each nonempty overlap, such that the ''cocycle condition'' :

t_(x) = t_(x)t_(x) \qquad \forall x \in U_i \cap U_j \cap U_k

holds, there exists a fiber bundle ''E'' → ''X'' with fiber ''F'' and structure group ''G'' that is trivializable over with transition functions ''t''_''ij''. Let ''E''′ be another fiber bundle with the same base space, fiber, structure group, and trivializing neighborhoods, but transition functions ''t''′_''ij''. If the action of ''G'' on ''F'' is faithful, then ''E''′ and ''E'' are isomorphic

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

there exist functions :

t_i : U_i \to G

such that :

t'_(x) = t_i(x)^t_(x)t_j(x) \qquad \forall x \in U_i \cap U_j.

Taking ''t''_''i'' to be constant functions to the identity in ''G'', we see that two fiber bundles with the same base, fiber, structure group, trivializing neighborhoods, and transition functions are isomorphic. A similar theorem holds in the smooth category, where ''X'' and ''Y'' are

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

s, ''G'' is a

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...

with a smooth left action on ''Y'' and the maps ''t''_''ij'' are all smooth.

Construction

The proof of the theorem is

constructive Although the general English usage of the adjective constructive is "helping to develop or improve something; helpful to someone, instead of upsetting and negative," as in the phrase "constructive criticism," in legal writing ''constructive'' has ...

. That is, it actually constructs a fiber bundle with the given properties. One starts by taking the

disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A (th ...

of the

product space In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...

s ''U''_''i'' × ''F'' :

T = \coprod_U_i \times F = \

and then forms the

quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...

by the

equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...

(j,x,y) \sim (i,x,t_(x)\cdot y)\qquad \forall x\in U_i \cap U_j, y\in F.

The total space ''E'' of the bundle is ''T''/~ and the projection π : ''E'' → ''X'' is the map which sends the equivalence class of (''i'', ''x'', ''y'') to ''x''. The local trivializations :

\phi_i : \pi^(U_i) \to U_i \times F

are then defined by :

\phi_i^(x,y) = i,x,y)

Associated bundle

Let ''E'' → ''X'' a fiber bundle with fiber ''F'' and structure group ''G'', and let ''F''′ be another left ''G''-space. One can form an associated bundle ''E''′ → ''X'' with a fiber ''F''′ and structure group ''G'' by taking any local trivialization of ''E'' and replacing ''F'' by ''F''′ in the construction theorem. If one takes ''F''′ to be ''G'' with the action of left multiplication then one obtains the associated

principal bundle In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equip ...

References

* *{{cite book , last = Steenrod , first = Norman , title = The Topology of Fibre Bundles , url = https://archive.org/details/topologyoffibreb0000stee , url-access = registration , publisher = Princeton University Press , location = Princeton , year = 1951 , isbn = 0-691-00548-6 See Part I, §2.10 and §3. Fiber bundles Theorems in topology