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.^[1]

A Pinched Torus

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:

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

Topology

Topologically, the pinched torus is homotopy equivalent to the wedge of a sphere and a circle.^[2][3] It is homeomorphic to a sphere with two distinct points being identified.^[2][3]

Homology

Let P denote the pinched torus. The homology groups of P over the integers can be calculated. They are given by:

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

Cohomology

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

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