Pinched torus

Short description: Kind of two-dimensional surface

A Pinched Torus

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.

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:

$f(x,y)=(g(x,y)\cos x,g(x,y)\sin x,\sin \left(\frac{x}{2}\right)\sin y)$

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:

$H_0(P,\mathbb{\mathbb{Z}})\cong \mathbb{\mathbb{Z}},H_1(P,\mathbb{\mathbb{Z}})\cong \mathbb{\mathbb{Z}},\text{and}{H}_2(P,\mathbb{\mathbb{Z}})\cong \mathbb{\mathbb{Z}}.$

Cohomology

The cohomology groups of P over the integers can be calculated. They are given by:

$H^0(P,\mathbb{\mathbb{Z}})\cong \mathbb{\mathbb{Z}},H^1(P,\mathbb{\mathbb{Z}})\cong \mathbb{\mathbb{Z}},\text{and}{H}^2(P,\mathbb{\mathbb{Z}})\cong \mathbb{\mathbb{Z}}.$