The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.

Fibrations are used, for example, in Postnikov systems or obstruction theory.

In this article, all mappings are continuous mappings between topological spaces.

Formal definitions

Homotopy lifting property

A mapping ${\displaystyle p\colon E\to B}$ satisfies the homotopy lifting property for a space ${\displaystyle X}$ if:

• for every homotopy ${\displaystyle h\colon X\times \to B}$ and
• for every mapping (also called lift) ${\displaystyle {\tilde {h}}_{0}\colon X\to E}$ lifting ${\displaystyle h|_{X\times 0}=h_{0}}$ (i.e. ${\displaystyle h_{0}=p\circ {\tilde {h}}_{0}}$)

there exists a (not necessarily unique) homotopy ${\displaystyle {\tilde {h}}\colon X\times \to E}$ lifting ${\displaystyle h}$ (i.e. ${\displaystyle h=p\circ {\tilde {h}}}$) with ${\displaystyle {\tilde {h}}_{0}={\tilde {h}}|_{X\times 0}.}$

The following commutative diagram shows the situation: ^[1]: 66

Fibration

A fibration (also called Hurewicz fibration) is a mapping ${\displaystyle p\colon E\to B}$ satisfying the homotopy lifting property for all spaces ${\displaystyle X.}$ The space ${\displaystyle B}$ is called base space and the space ${\displaystyle E}$ is called total space. The fiber over ${\displaystyle b\in B}$ is the subspace ${\displaystyle F_{b}=p^{-1}(b)\subseteq E.}$^[1]: 66

Serre fibration

A Serre fibration (also called weak fibration) is a mapping ${\displaystyle p\colon E\to B}$ satisfying the homotopy lifting property for all CW-complexes.^{[2]: 375-376}

Every Hurewicz fibration is a Serre fibration.

Quasifibration

A mapping ${\displaystyle p\colon E\to B}$ is called quasifibration, if for every ${\displaystyle b\in B,}$ ${\displaystyle e\in p^{-1}(b)}$ and ${\displaystyle i\geq 0}$ holds that the induced mapping ${\displaystyle p_{*}\colon \pi _{i}(E,p^{-1}(b),e)\to \pi _{i}(B,b)}$ is an isomorphism.

Every Serre fibration is a quasifibration.^{[3]: 241-242}

Examples

• The projection onto the first factor ${\displaystyle p\colon B\times F\to B}$ is a fibration. That is, trivial bundles are fibrations.
• Every covering ${\displaystyle p\colon E\to B}$ is a fibration. Specifically, for every homotopy ${\displaystyle h\colon X\times \to B}$ and every lift ${\displaystyle {\tilde {h}}_{0}\colon X\to E}$ there exists a uniquely defined lift ${\displaystyle {\tilde {h}}\colon X\times \to E}$ with ${\displaystyle p\circ {\tilde {h}}=h.}$^{[4]: 159  [5]: 50}
• Every fiber bundle ${\displaystyle p\colon E\to B}$ satisfies the homotopy lifting property for every CW-complex.^[2]: 379
• A fiber bundle with a paracompact and Hausdorff base space satisfies the homotopy lifting property for all spaces.^[2]: 379
• An example of a fibration which is not a fiber bundle is given by the mapping ${\displaystyle i^{*}\colon X^{I^{k}}\to X^{\partial I^{k}}}$ induced by the inclusion $$Zdroj:https://en.wikipedia.org?pojem=Fibration
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