Sections
A section (or cross section) of a fiber bundle is a continuous map f : B → E such that π(f(x))=x for all x in B. Since bundles do not in general have globally defined sections, one of the purposes of the theory is to account for their existence. The obstruction to the existence of a section can often be measured by a cohomology class, which leads to the theory of characteristic classes in algebraic topology.
The most well-known example is the hairy ball theorem, where the Euler class is the obstruction to the tangent bundle of the 2-sphere having a nowhere vanishing section.
Often one would like to define sections only locally (especially when global sections do not exist). A local section of a fiber bundle is a continuous map f : U → E where U is an open set in B and π(f(x))=x for all x in U. If (U, φ) is a local trivialization chart then local sections always exist over U. Such sections are in 1-1 correspondence with continuous maps U → F. Sections form a sheaf.
Read more about this topic: Fiber Bundle
Famous quotes containing the word sections:
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—Gaston Bachelard (1884–1962)
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—William Howard Taft (1857–1930)
“I have a new method of poetry. All you got to do is look over your notebooks ... or lay down on a couch, and think of anything that comes into your head, especially the miseries.... Then arrange in lines of two, three or four words each, don’t bother about sentences, in sections of two, three or four lines each.”
—Allen Ginsberg (b. 1926)