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Fibration. 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 [1] : 66.
Fiber (mathematics) In mathematics, the fiber ( US English) or fibre ( British English) of an element under a function is the preimage of the singleton set , [1] : p.69 that is. As an example of abuse of notation, this set is often denoted as , which is technically incorrect since the inverse relation of is not necessarily a function.
Pullback bundle. In mathematics, a pullback bundle or induced bundle [1] [2] [3] 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 ...
Homotopy fiber. In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber) [1] is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces . It acts as a homotopy theoretic kernel of a mapping of topological spaces due to the fact it yields a long exact ...
Path space fibration. In algebraic topology, the path space fibration over a based space [1] is a fibration of the form [2] where. is the based path space of X; that is, equipped with the compact-open topology. is the fiber of over the base point of X; thus it is the loop space of X. The free path space of X, that is, , consists of all maps ...
Bundle map. In mathematics, a bundle map (or bundle morphism) 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 ...
In mathematics, and particularly topology, a fiber bundle ( Commonwealth English: fibre 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 ...
Gysin homomorphism. In the field of mathematics known as algebraic topology, the Gysin sequence is a long exact sequence which relates the cohomology classes of the base space, the fiber and the total space of a sphere bundle. The Gysin sequence is a useful tool for calculating the cohomology rings given the Euler class of the sphere bundle and ...