Orbifold

In topology and group theory, an orbifold is a generalization of a manifold.

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It is a topological space (called an underlying space) with an orbifold structure (see below).

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The underlying space locally looks like a quotient of a

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Euclidean space under the action of a finite group of isometries.

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In string theory, the word "orbifold" has additional meaning, discussed below.

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The main example of underlying space is a quotient space of a manifold under the action of a finite group of diffeomorphisms, in particular a manifold with boundary carries natural

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orbifold structure, since it is Z2-factor of its

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double.

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A factor space of a manifold along a smooth S^1-action without fixed points carries the structure of an orbifold (this is not a partial case of the main example).

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Orbifold structure gives a natural stratification by open manifolds on its underlying space, where one strata corresponds to a set of singular points of the same type.

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It should be noted that one topological space can carry many different

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orbifold structures.

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For example, consider the orbifold O associated with a

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factor space of 2-sphere along a rotation by pi^{}_{} ,

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it is homeomorphic to 2-sphere, but the natural orbifold structure is different.

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It is possible to adopt most of the characteristics of manifolds to orbifolds and

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these characteristics are usually different from correspondent characteristics of underlying space.

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In the above example, its orbifold fundamental group of O is Z2

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and its orbifold Euler characteristic is 1.

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Topology: Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. When the discipline was first introduced it was called analysis situs (Latin analysis of place)....

Group theory: Group theory is that branch of mathematics concerned with the study of groups....

Euclidean space: In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. The generalization applies Euclid's concept of distance, and the related concepts of length and angle, to a coordinate system in any number of dimensions. It is the "standard" example of a finit...