Geometric topology | General topology | Homeomorphisms

Mapping torus

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: The result is a fiber bundle whose base is a circle and whose fiber is the original space X. If X is a manifold, Mf will be a manifold of dimension one higher, and it is said to "fiber over the circle". As a simple example, let be the circle, and be the inversion , then the mapping torus is the Klein bottle. 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 Mf 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 Mf is hyperbolic if and only if f is a pseudo-Anosov homeomorphism of S. (Wikipedia).

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Torus Magic

Buy at http://www.shapeways.com/shops/GeometricToy Torus Magic is a transformable torus. This torus object is constructed with many rings,and transforms flat,spherical etc. Also you can turn inside out the torus. Copyright (c) 2014,AkiraNishihara

From playlist 3D printed toys

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Torus Earth

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From playlist 3D printing

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360-Degree Tour INSIDE a TORUS with GeoGebra Augmented Reality

Come take a virtual tour as we walk around INSIDE a closed torus! Also, come explore some lesson ideas with respect to using GeoGebra Augmented Reality to explore conic sections, form surfaces of revolutions (by rotating graphs about the x-axis & y-axis), and model in 3D! https://www.geoge

From playlist GeoGebra Augmented Reality (older iOS app)

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Torus Magic (50mm)

Buy at http://www.shapeways.com/shops/GeometricToy Torus Magic is a transformable torus. This torus object is constructed with 20 large rings(50mm diameter) and many small rings.It transforms flat,spherical etc. Also you can turn inside out the torus. Copyright (c) 2015,AkiraNishihara

From playlist 3D printed toys

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Morphing a Torus: Quick GeoGebra 3D Graphing Calculator with Augmented Reality Demo

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From playlist GeoGebra 3D with AR (iOS): Explorations, Demos, and Lesson Ideas

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Torus Magic with Ring 1

Buy at http://www.shapeways.com/shops/GeometricToy "Torus Magic" can eat another torus.This torus object is constructed with 30 large rings(70mm diameter) and many small rings. Copyright (c) 2015,AkiraNishihara

From playlist 3D printed toys

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From playlist IMPRS Ringvorlesung - Introduction to Nonlinear Algebra

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A Continuous Transformation of a Double Cover of the Complex Plane into a Torus

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From playlist IMPRS Ringvorlesung - Introduction to Nonlinear Algebra

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Random walks on Tori and normal numbers in self similar sets by Arijit Ganguly

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From playlist Smooth And Homogeneous Dynamics

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Entropy, Algebraic Integers and Moduli of Surfaces - Curtis McMullen

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From playlist Mathematics

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Mirror symmetry and cluster algebras – Paul Hacking & Sean Keel – ICM2018

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From playlist Algebraic & Complex Geometry

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Dimers and Beauville Integrable systems by Terrence George

PROGRAM: COMBINATORIAL ALGEBRAIC GEOMETRY: TROPICAL AND REAL (HYBRID) ORGANIZERS: Arvind Ayyer (IISc, India), Madhusudan Manjunath (IITB, India) and Pranav Pandit (ICTS-TIFR, India) DATE: 27 June 2022 to 08 July 2022 VENUE: Madhava Lecture Hall and Online Algebraic geometry is the study

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From playlist Mathematics

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From playlist Mathematics

Related pages

Manifold | Topological space | Mathematics | Fiber bundle | Homeomorphism | Pseudo-Anosov map | 3-manifold | Topology | Surface (topology) | Hyperbolic manifold | William Thurston | Cartesian product | Circle | Closed manifold