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In themathematical field oftopology, asection (orcross section)^[1] of afiber bundle${\displaystyle E}$ is a continuousright inverse of theprojection function${\displaystyle \pi }$. In other words, if${\displaystyle E}$ is a fiber bundle over abase space,${\displaystyle B}$:

${\displaystyle \pi :E\to B}$

then a section of that fiber bundle is acontinuous map,

${\displaystyle \sigma :B\to E}$

such that

${\displaystyle \pi (\sigma (x))=x}$ for all${\displaystyle x\in B}$.

A section is an abstract characterization of what it means to be agraph. The graph of a function${\displaystyle g:B\to Y}$ can be identified with a function taking its values in theCartesian product${\displaystyle E=B\times Y}$, of${\displaystyle B}$ and${\displaystyle Y}$:

${\displaystyle \sigma :B\to E,\sigma (x)=(x,g(x))\in E.}$

Let${\displaystyle \pi :E\to B}$ be the projection onto the first factor:${\displaystyle \pi (x,y)=x}$. Then a graph is any function${\displaystyle \sigma }$ for which${\displaystyle \pi (\sigma (x))=x}$.

The language of fibre bundles allows this notion of a section to be generalized to the case when${\displaystyle E}$ is not necessarily a Cartesian product. If${\displaystyle \pi :E\to B}$ is a fibre bundle, then a section is a choice of point${\displaystyle \sigma (x)}$ in each of the fibres. The condition${\displaystyle \pi (\sigma (x))=x}$ simply means that the section at a point${\displaystyle x}$ must lie over${\displaystyle x}$. (See image.)

For example, when${\displaystyle E}$ is avector bundle a section of${\displaystyle E}$ is an element of the vector space${\displaystyle E_x}$ lying over each point${\displaystyle x\in B}$. In particular, avector field on asmooth manifold${\displaystyle M}$ is a choice oftangent vector at each point of${\displaystyle M}$: this is asection of thetangent bundle of${\displaystyle M}$. Likewise, a1-form on${\displaystyle M}$ is a section of thecotangent bundle.

Sections, particularly ofprincipal bundles and vector bundles, are also very important tools indifferential geometry. In this setting, the base space${\displaystyle B}$ is asmooth manifold${\displaystyle M}$, and${\displaystyle E}$ is assumed to be a smooth fiber bundle over${\displaystyle M}$ (i.e.,${\displaystyle E}$ is a smooth manifold and${\displaystyle \pi :E\to M}$ is asmooth map). In this case, one considers the space ofsmooth sections of${\displaystyle E}$ over an open set${\displaystyle U}$, denoted${\displaystyle C^{\mathrm{\infty }}(U,E)}$. It is also useful ingeometric analysis to consider spaces of sections with intermediate regularity (e.g.,${\displaystyle C^k}$ sections, or sections with regularity in the sense ofHölder conditions orSobolev spaces).

Fiber bundles do not in general have suchglobal sections (consider, for example, the fiber bundle over${\displaystyle S^1}$ with fiber${\displaystyle F=\mathbb{R}\setminus \{0\}}$ obtained by taking theMöbius bundle and removing the zero section), so it is also useful to define sections only locally. Alocal section of a fiber bundle is a continuous map${\displaystyle s:U\to E}$ where${\displaystyle U}$ is anopen set in${\displaystyle B}$ and${\displaystyle \pi (s(x))=x}$ for all${\displaystyle x}$ in${\displaystyle U}$. If${\displaystyle (U,\phi )}$ is alocal trivialization of${\displaystyle E}$, where${\displaystyle \phi }$ is a homeomorphism from${\displaystyle {\pi }^{-1}(U)}$ to${\displaystyle U\times F}$ (where${\displaystyle F}$ is thefiber), then local sections always exist over${\displaystyle U}$ in bijective correspondence with continuous maps from${\displaystyle U}$ to${\displaystyle F}$. The (local) sections form asheaf over${\displaystyle B}$ called thesheaf of sections of${\displaystyle E}$.

The space of continuous sections of a fiber bundle${\displaystyle E}$ over${\displaystyle U}$ is sometimes denoted${\displaystyle C(U,E)}$, while the space of global sections of${\displaystyle E}$ is often denoted${\displaystyle \mathrm{\Gamma }(E)}$ or${\displaystyle \mathrm{\Gamma }(B,E)}$.

Sections are studied inhomotopy theory andalgebraic topology, where one of the main goals is to account for the existence or non-existence ofglobal sections. Anobstruction 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 particularcharacteristic classes, which are cohomological classes. For example, aprincipal bundle has a global section if and only if it istrivial. On the other hand, avector bundle always has a global section, namely thezero section. However, it only admits a nowhere vanishing section if itsEuler class is zero.

Obstructions to extending local sections may be generalized in the following manner: take atopological space and form acategory 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 ofabelian 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 afixed vector space is assigned. However, sheaves can "continuously change" the vector space (or more generally abelian group).

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

• Section (category theory)
• Fibration
• Gauge theory (mathematics)
• Principal bundle
• Pullback bundle
• Vector bundle

• 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.

• Fiber bundles
• Differential topology
• Algebraic topology
• Homotopy theory

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