In the mathematical field of topology, a **section** (or **cross section**)^{ [1] } of a fiber bundle is a continuous right inverse of the projection function . In other words, if is a fiber bundle over a base space, :

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

such that

- for all .

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

Let be the projection onto the first factor: . Then a graph is any function for which .

The language of fibre bundles allows this notion of a section to be generalized to the case when is not necessarily a Cartesian product. If is a fibre bundle, then a section is a choice of point in each of the fibres. The condition simply means that the section at a point must lie over . (See image.)

For example, when is a vector bundle a section of is an element of the vector space lying over each point . In particular, a vector field on a smooth manifold is a choice of tangent vector at each point of : this is a *section* of the tangent bundle of . Likewise, a 1-form on 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 is a smooth manifold , and is assumed to be a smooth fiber bundle over (i.e., is a smooth manifold and is a smooth map). In this case, one considers the space of **smooth sections** of over an open set , denoted . It is also useful in geometric analysis to consider spaces of sections with intermediate regularity (e.g., sections, or sections with regularity in the sense of Hölder conditions or Sobolev spaces).

Fiber bundles do not in general have such *global* sections (consider, for example, the fiber bundle over with fiber 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 where is an open set in and for all in . If is a local trivialization of , where is a homeomorphism from to (where is the fiber), then local sections always exist over in bijective correspondence with continuous maps from to . The (local) sections form a sheaf over called the **sheaf of sections** of .

The space of continuous sections of a fiber bundle over is sometimes denoted , while the space of global sections of is often denoted or .

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.

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.

- ↑ Husemöller, Dale (1994),
*Fibre Bundles*, Springer Verlag, p. 12, ISBN 0-387-94087-1

In mathematics, **orientability** is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is **orientable** if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an **orientation** of the space. Real vector spaces, Euclidean spaces, and spheres are orientable. A space is **non-orientable** if "clockwise" is changed into "counterclockwise" after running through some loops in it, and coming back to the starting point. This means that a Geometric shape, such as , that moves continuously along such a loop is changed in its own mirror image . A Möbius strip is an example of a non-orientable space.

In mathematics, especially differential geometry, the **cotangent bundle** of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may be generalized to categories with more structure than smooth manifolds, such as complex manifolds, or algebraic varieties or schemes. In the smooth case, any Riemannian metric or symplectic form gives an isomorphism between the cotangent bundle and the tangent bundle, but they are not in general isomorphic in other categories.

In mathematics, a **sheaf** is a tool for systematically tracking data attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set.

In mathematics, a **vector bundle** is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space *X* : to every point *x* of the space *X* we associate a vector space *V*(*x*) in such a way that these vector spaces fit together to form another space of the same kind as *X*, which is then called a **vector bundle over X**.

In mathematics, and particularly topology, a **fiber bundle** is a space that is *locally* a product space, but *globally* may have a different topological structure. Specifically, the similarity between a space and a product space is defined using a continuous surjective map, , that in small regions of behaves just like a projection from corresponding regions of to . The map , called the **projection** or **submersion** of the bundle, is regarded as part of the structure of the bundle. The space is known as the **total space** of the fiber bundle, as the **base space**, and the **fiber**.

In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the **Chern classes** are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau manifolds, string theory, Chern–Simons theory, knot theory, Gromov–Witten invariants, topological quantum field theory, the Chern theorem etc.

In mathematics, a **principal bundle** is a mathematical object that formalizes some of the essential features of the Cartesian product of a space with a group . In the same way as with the Cartesian product, a principal bundle is equipped with

- An action of on , analogous to for a product space.
- A projection onto . For a product space, this is just the projection onto the first factor, .

In mathematics, especially in algebraic geometry and the theory of complex manifolds, **coherent sheaves** are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information.

In differential topology, the **jet bundle** is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. Jets may also be seen as the coordinate free versions of Taylor expansions.

In mathematics, a **pullback bundle** or **induced bundle** is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle *π* : *E* → *B* and a continuous map *f* : *B*′ → *B* one can define a "pullback" of *E* by *f* as a bundle *f*^{*}*E* over *B*′. The fiber of *f*^{*}*E* over a point *b*′ in *B*′ is just the fiber of *E* over *f*(*b*′). Thus *f*^{*}*E* is the disjoint union of all these fibers equipped with a suitable topology.

In differential geometry, in the category of differentiable manifolds, a **fibered manifold** is a surjective submersion

In mathematics, **local coefficients** is an idea from algebraic topology, a kind of half-way stage between homology theory or cohomology theory with coefficients in the usual sense, in a fixed abelian group *A*, and general sheaf cohomology which, roughly speaking, allows coefficients to vary from point to point in a topological space *X*. Such a concept was introduced by Norman Steenrod in 1943.

In mathematics, the **Puppe sequence** is a construction of homotopy theory, so named after Dieter Puppe. It comes in two forms: a long exact sequence, built from the mapping fibre, and a long coexact sequence, built from the mapping cone. Intuitively, the Puppe sequence allows us to think of homology theory as a functor that takes spaces to long-exact sequences of groups. It is also useful as a tool to build long exact sequences of relative homotopy groups.

In differential geometry, an **Ehresmann connection** is a version of the notion of a connection, which makes sense on any smooth fiber bundle. In particular, it does not rely on the possible vector bundle structure of the underlying fiber bundle, but nevertheless, linear connections may be viewed as a special case. Another important special case of Ehresmann connections are principal connections on principal bundles, which are required to be equivariant in the principal Lie group action.

In mathematics, a **bundle map** is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There are also several variations on the basic theme, depending on precisely which category of fiber bundles is under consideration. In the first three sections, we will consider general fiber bundles in the category of topological spaces. Then in the fourth section, some other examples will be given.

In mathematics, especially in algebraic geometry and the theory of complex manifolds, **coherent sheaf cohomology** is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the existence of sections of line bundles or of more general coherent sheaves; such sections can be viewed as generalized functions. Cohomology provides computable tools for producing sections, or explaining why they do not exist. It also provides invariants to distinguish one algebraic variety from another.

In mathematics, a **projective bundle** is a fiber bundle whose fibers are projective spaces.

This is a **glossary of algebraic geometry**.

In mathematics, given an action of a group scheme *G* on a scheme *X* over a base scheme *S*, an **equivariant sheaf***F* on *X* is a sheaf of -modules together with the isomorphism of -modules

This is a glossary of properties and concepts in algebraic topology in mathematics.

- Norman Steenrod,
*The Topology of Fibre Bundles*, Princeton University Press (1951). ISBN 0-691-00548-6. - David Bleecker,
*Gauge Theory and Variational Principles*, Addison-Wesley publishing, Reading, Mass (1981). ISBN 0-201-10096-7. - Husemöller, Dale (1994),
*Fibre Bundles*, Springer Verlag, ISBN 0-387-94087-1

- Fiber Bundle, PlanetMath
- Weisstein, Eric W. "Fiber Bundle".
*MathWorld*.

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