Pinched torus explained

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]

Parametrisation

A pinched torus is easily parametrisable. Let us write . An example of such a parametrisation − which was used to plot the picture − is given by where:

f(x,y) = \left( g(x,y)\cos x , g(x,y)\sin x , \sin\left( (x)/(2) \right)\sin y \right)

Topology

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

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,\Z)} \cong \Z, H _{1(P,\Z)} \cong \Z, and H _{2(P,\Z)} \cong \Z.

Cohomology

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

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

Notes and References

1. Brasselet. J. P.. 1996 . Intersection of Algebraic Cycles . Journal of Mathematical Sciences. Springer New York. 82. 5. 3625–3632. 10.1007/bf02362566.
2. Web site: Chapter 0: Algebraic Topology. Allen Hatcher. August 6, 2010.