Bundle

Stratification - continuous surjective mapping

\pi \colon X\to B

{\ displaystyle \ pi \ colon X \ to B}

\ pi \ colon X \ to B

between topological spaces .

Wherein

$X$ ${\ displaystyle X}$ $X$ called the bundle space (or the total bundle space or the bundle space )
$B$ ${\ displaystyle B}$ $B$ - the base of the bundle,
$\pi$ ${\ displaystyle \ pi}$ $\ pi$ - projection of the bundle,
$F_{b}=\pi ^{-1}(b)$ ${\ displaystyle F_ {b} = \ pi ^ {- 1} (b)}$ $F_ {b} = \ pi ^ {{- 1}} (b)$ - layer above $b\in B$ ${\ displaystyle b \ in B}$ $b \ in B$ .

Usually, a bundle is represented as a union of layers $F_{b}$ ${\ displaystyle F_ {b}}$ $F_ {b}$ parameterized by the base $B$ ${\ displaystyle B}$ $B$ and glued by the topology of space $X$ ${\ displaystyle X}$ $X$ .

Often the term “bundle” is used as a short name for more specific terms, such as smooth bundle or locally trivial bundle .

Related Definitions

Bundle section $\pi :X\to B$ ${\ displaystyle \ pi: X \ to B}$ mapping $h:B\to X$ ${\ displaystyle h: B \ to X}$ such that $\pi \circ h$ ${\ displaystyle \ pi \ circ h}$ - identity mapping onto $B$ ${\ displaystyle B}$ .
A bundle is called trivial if its space is homeomorphic to the direct product $B\times F$ ${\ displaystyle B \ times F}$ , and the projection is defined in a canonical way:

\pi (b,f)=b,\quad b\in B,~f\in F

{\ displaystyle \ pi (b, f) = b, \ quad b \ in B, ~ f \ in F}

Types of bundles

Locally trivial bundle
Gurevich bundle
Seifert bundle
Bundle serre
Hopf bundle
Smooth bundle
Vector bundle

Literature

Vasiliev V.A. Introduction to topology. - M .: FAZIS, 1997 .-- 132 p. - ISBN 5-7036-0036-7 .

Rokhlin V.A., Fuchs D.B. Beginner course in topology. Geometric chapters. - M .: Nauka, 1977 .-- 487 p.

Kobayashi Sh., Nomizu K. Fundamentals of differential geometry, v. 1. - M .: Nauka, 1981 .-- 344 p.

Bundle

Related Definitions

Types of bundles

Literature

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