Section (fiber bundle)

In the mathematical field of topology, a section (or cross section) of a fiber bundle formula_1 is a continuous right inverse of the projection function formula_2. In other words, if formula_1 is a fiber bundle over a base space, formula_4:
then a section of that fiber bundle is a continuous map,
such that
A section is an abstract characterization of what it means to be a graph. The graph of a function formula_9 can be identified with a function taking its values in the Cartesian product formula_10, of formula_11 and formula_12:
Let formula_14 be the projection onto the first factor: formula_15. Then a graph is any function formula_16 for which formula_7.
The language of fibre bundles allows this notion of a section to be generalized to the case when formula_1 is not necessarily a Cartesian product. If formula_14 is a fibre bundle, then a section is a choice of point formula_20 in each of the fibres. The condition formula_7 simply means that the section at a point formula_22 must lie over formula_22. (See image.)
For example, when formula_1 is a vector bundle a section of formula_1 is an element of the vector space formula_26 lying over each point formula_27. In particular, a vector field on a smooth manifold formula_28 is a choice of tangent vector at each point of formula_28: this is a "section" of the tangent bundle of formula_28. Likewise, a 1-form on formula_28 is a section of the cotangent bundle.
Sections, particularly of principal bundles and vector bundles, are also very important tools in differential geometry. In this setting, the base space formula_4 is a smooth manifold formula_28, and formula_1 is assumed to be a smooth fiber bundle over formula_28 (i.e., formula_1 is a smooth manifold and formula_37 is a smooth map). In this case, one considers the space of smooth sections of formula_1 over an open set formula_39, denoted formula_40. It is also useful in geometric analysis to consider spaces of sections with intermediate regularity (e.g., formula_41 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" sections (consider, for example, the fiber bundle over formula_42 with fiber formula_43 obtained by taking the Möbius bundle and removing the zero section), so it is also useful to define sections only locally. A local section of a fiber bundle is a continuous map formula_44 where formula_39 is an open set in formula_4 and formula_47 for all formula_48 in formula_39. If formula_50 is a local trivialization of formula_1, where formula_52 is a homeomorphism from formula_53 to formula_54 (where formula_55 is the fiber), then local sections always exist over formula_39 in bijective correspondence with continuous maps from formula_39 to formula_55. The (local) sections form a sheaf over formula_4 called the sheaf of sections of formula_1.
The space of continuous sections of a fiber bundle formula_1 over formula_39 is sometimes denoted formula_63, while the space of global sections of formula_1 is often denoted formula_65 or formula_66.
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, it only admits a nowhere vanishing section if its Euler class is zero.
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.