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In mathematics, and especially topology and differential geometry, a pinched torus (or croissant surface) is a kind of two-dimensional surface. It gets its name from its resemblance to a torus that has been pinched at a single point. A pinched torus is an example of an orientable, compact 2-dimensional pseudomanifold.
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Parametrisation
A pinched torus is easily parametrisable. Let us write g(x,y) = 2 + sin(x/2).cos(y). An example of such a parametrisation − which was used to plot the picture − is given by ƒ : [0,2π)2 → R3 where:
Topology
Topologically, the pinched torus is homotopy equivalent to the wedge of a sphere and a circle. It is homeomorphic to a sphere with two distinct points being identified.
Homology
Let P denote the pinched torus. The homology groups of P over the integers can be calculated. They are given by:
Cohomology
The cohomology groups of P over the integers can be calculated. They are given by: