HOME

TheInfoList



OR:

In
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 ...
, a bundle map (or bundle morphism) is a
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
in the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
of
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 ...
s. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common
base space 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 an ...
. There are also several variations on the basic theme, depending on precisely which category of fiber bundles is under consideration. In the first three sections, we will consider general fiber bundles in the
category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again cont ...
. Then in the fourth section, some other examples will be given.


Bundle maps over a common base

Let \pi_E\colon E \to M and \pi_F\colon F \to M be fiber bundles over a space ''M''. Then a bundle map from ''E'' to ''F'' over ''M'' is a continuous map \varphi\colon E \to F such that \pi_F\circ\varphi = \pi_E . That is, the diagram should
commute Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...
. Equivalently, for any point ''x'' in ''M'', \varphi maps the fiber E_x= \pi_E^(\) of ''E'' over ''x'' to the fiber F_x= \pi_F^(\) of ''F'' over ''x''.


General morphisms of fiber bundles

Let π''E'':''E''→ ''M'' and π''F'':''F''→ ''N'' be fiber bundles over spaces ''M'' and ''N'' respectively. Then a continuous map \varphi : E \to F is called a bundle map from ''E'' to ''F'' if there is a continuous map ''f'':''M''→ ''N'' such that the diagram commutes, that is, \pi_F\circ\varphi = f\circ\pi_E . In other words, \varphi is fiber-preserving, and ''f'' is the induced map on the space of fibers of ''E'': since π''E'' is surjective, ''f'' is uniquely determined by \varphi. For a given ''f'', such a bundle map \varphi is said to be a bundle map ''covering f''.


Relation between the two notions

It follows immediately from the definitions that a bundle map over ''M'' (in the first sense) is the same thing as a bundle map covering the identity map of ''M''. Conversely, general bundle maps can be reduced to bundle maps over a fixed base space using the notion of a pullback bundle. If π''F'':''F''→ ''N'' is a fiber bundle over ''N'' and ''f'':''M''→ ''N'' is a continuous map, then the pullback of ''F'' by ''f'' is a fiber bundle ''f''*''F'' over ''M'' whose fiber over ''x'' is given by (''f''*''F'')''x'' = ''F''''f''(''x''). It then follows that a bundle map from ''E'' to ''F'' covering ''f'' is the same thing as a bundle map from ''E'' to ''f''*''F'' over ''M''.


Variants and generalizations

There are two kinds of variation of the general notion of a bundle map. First, one can consider fiber bundles in a different category of spaces. This leads, for example, to the notion of a smooth bundle map between smooth fiber bundles over a
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 ...
. Second, one can consider fiber bundles with extra structure in their fibers, and restrict attention to bundle maps which preserve this structure. This leads, for example, to the notion of a (vector) bundle homomorphism between
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
s, in which the fibers are vector spaces, and a bundle map ''φ'' is required to be a linear map on each fiber. In this case, such a bundle map ''φ'' (covering ''f'') may also be viewed as a
section Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...
of the vector bundle Hom(''E'',''f*F'') over ''M'', whose fiber over ''x'' is the vector space Hom(''Ex'',''F''''f''(''x'')) (also denoted ''L''(''Ex'',''F''''f''(''x''))) of
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
s from ''Ex'' to ''F''''f''(''x''). {{DEFAULTSORT:Bundle Map Fiber bundles Theory of continuous functions