Section (fiber bundle)

Cuts are pictures which are in the algebraic topology, in particular in the homotopy examined. In particular, one is interested in the conditions under which such maps exist. Probably the best known example of cuts are the differential forms.

Motivation

A section can be thought of as a generalization of the graph of a function. The graph of an image can be identified with a function with values ​​in the Cartesian product. The function has the form

Is the projection on the first component, the following applies. As will be shown, the following definition, is a special case of a cut.

With the help of sections in fiber bundles can be above construction also generalizes to amounts which do not consist of Cartesian products.

Definition

Section

Whether it is a fiber bundle consisting of the total area, the base area, the projection beam and the fiber. A ( global ) section in a fiber bundle is a continuous map such that

Applies to all. The picture is therefore a right inverse to the bundle projection. The set of (global) sections is often referred to with or with.

Section with compact support

Whether it is a fiber bundle. A cut is called section with compact support if there is a compact set with for. The set of sections with compact support is denoted by or with. Instead of addition, the addition is used.

Smooth cut

Is a smooth manifold, a smooth vector bundle over and is the picture from the above section smooth, it is called a smooth (global) section. To distinguish it from the previously defined sections you quoted this amount of these cuts means. Can not create confusion between smooth and non-smooth cuts occur, we often waived again on the addition.

Examples

Local section

General fiber bundles, in contrast to the above examples do not always global sections. Therefore, it seems reasonable to define cuts locally.

Be an open subset. A local interface in a fiber bundle is an image for which also applies to all.