Mapping torus

Last updated April 21, 2019

In mathematics, the mapping torus in topology of a homeomorphism f of some topological space X to itself is a particular geometric construction with f. Take the cartesian product of X with a closed interval I, and glue the boundary components together by the static homeomorphism:

Mathematics Field of study concerning quantity, patterns and change

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In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing.

Homeomorphism In mathematics, isomorphism of topological spaces

In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word homeomorphism comes from the Greek words ὅμοιος (homoios) = similar or same and μορφή (morphē) = shape, form, introduced to mathematics by Henri Poincaré in 1895.

M_{f}={\frac {(I\times X)}{(1,x)\sim (0,f(x))}}

The result is a fiber bundle whose base is a circle and whose fiber is the original space X.

Fiber bundle continuous surjection satisfying a local triviality condition

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If X is a manifold, M_f will be a manifold of dimension one higher, and it is said to "fiber over the circle".

Manifold topological space that at each point resembles Euclidean space

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. In this more precise terminology, a manifold is referred to as an n-manifold.

Mapping tori of surface homeomorphisms play a key role in the theory of 3-manifolds and have been intensely studied. If S is a closed surface of genus g ≥ 2 and if f is a self-homeomorphism of S, the mapping torus M_f is a closed 3-manifold that fibers over the circle with fiber S. A deep result of Thurston states that in this case the 3-manifold M_f is hyperbolic if and only if f is a pseudo-Anosov homeomorphism of S. [1]

In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.

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Circle simple curve of Euclidean geometry

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References

↑ W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bulletin of the American Mathematical Society, vol. 19 (1988), pp. 417–431

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[1] W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bulletin of the American Mathematical Society, vol. 19 (1988), pp. 417–431