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Wikipedia / Section (fiber bundle)

In the mathematical field of topology, a section (or cross section) of a fiber bundle is a continuous right inverse of the projection function is a fiber bundle over a base space:

then a section of that fiber bundle is a continuous map,

such that

  1. .

A section is an abstract characterization of what it means to be a graph can be identified with a function taking its values in the Cartesian product:

.

. (See image.)

is a vector bundle. In particular, a vector field on a smooth manifold is a choice of tangent vector: this is a section of the tangent bundle. Likewise, a 1-form is a section of the cotangent bundle.

Sections, particularly of principal bundles and vector bundles, are also very important tools in differential geometry is a smooth manifold is a smooth map). In this case, one considers the space of smooth sections. It is also useful in geometric analysis sections, or sections with regularity in the sense of Hölder conditions or Sobolev spaces).

Local and global sections

Fiber bundles do not in general have such global obtained by taking the Möbius bundle and removing the zero section), so it is also useful to define sections only locally. A local section is an open set is a local trivialization is the fiber. The (local) sections form a sheaf called the sheaf of sections.

.

Extending to global sections

Sections are studied in homotopy theory and algebraic topology, where one of the main goals is to account for the existence or non-existence of global sections. An obstruction denies the existence of global sections since the space is too "twisted". More precisely, obstructions "obstruct" the possibility of extending a local section to a global section due to the space's "twistedness". Obstructions are indicated by particular characteristic classes, which are cohomological classes. For example, a principal bundle has a global section if and only if it is trivial. On the other hand, a vector bundle always has a global section, namely the zero section. However, the Euler class obstructs the existence of a nowhere vanishing section.

Generalizations

Obstructions to extending local sections may be generalized in the following manner: take a topological space and form a category whose objects are open subsets, and morphisms are inclusions. Thus we use a category to generalize a topological space. We generalize the notion of a "local section" using sheaves of abelian groups, which assigns to each object an abelian group (analogous to local sections).

There is an important distinction here: intuitively, local sections are like "vector fields" on an open subset of a topological space. So at each point, an element of a fixed vector space is assigned. However, sheaves can "continuously change" the vector space (or more generally abelian group).

This entire process is really the global section functor, which assigns to each sheaf its global section. Then sheaf cohomology enables us to consider a similar extension problem while "continuously varying" the abelian group. The theory of characteristic classes generalizes the idea of obstructions to our extensions.

See also

Notes

References

  • Norman Steenrod, The Topology of Fibre Bundles, Princeton University Press (1951). .

  • David Bleecker, Gauge Theory and Variational Principles, Addison-Wesley publishing, Reading, Mass (1981). .