Fibration

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. A mapping satisfies the homotopy lifting property for a space if: for every homotopy and for every mapping (also called lift) lifting (i.e. ) there exists a (not necessarily unique) homotopy lifting (i.e. ) with The following commutative diagram shows the situation: A fibration (also called Hurewicz fibration) is a mapping satisfying the homotopy lifting property for all spaces The space is called base space and the space is called total space. The fiber over is the subspace A Serre fibration (also called weak fibration) is a mapping satisfying the homotopy lifting property for all CW-complexes. Every Hurewicz fibration is a Serre fibration. A mapping is called quasifibration, if for every and holds that the induced mapping is an isomorphism. Every Serre fibration is a quasifibration. The projection onto the first factor is a fibration. That is, trivial bundles are fibrations. Every covering satisfies the homotopy lifting property for all spaces. Specifically, for every homotopy and every lift there exists a uniquely defined lift with Every fiber bundle satisfies the homotopy lifting property for every CW-complex. A fiber bundle with a paracompact and Hausdorff base space satisfies the homotopy lifting property for all spaces. An example for a fibration, which is not a fiber bundle, is given by the mapping induced by the inclusion where a topological space and is the space of all continuous mappings with the compact-open topology. The Hopf fibration is a non trivial fiber bundle and specifically a Serre fibration.