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.[1]

## Contents

* 1 Parametrisation
* 2 Topology
* 2.1 Homology
* 2.2 Cohomology
* 3 References

## 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⁡(x2)sin⁡y)

## 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:

H0(P,ℤ)≅ℤ, H1(P,ℤ)≅ℤ, and H2(P,ℤ)≅ℤ.

### Cohomology

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

H0(P,ℤ)≅ℤ, H1(P,ℤ)≅ℤ, and H2(P,ℤ)≅ℤ.

## References

1. ↑ Brasselet, J. P. (1996). "Intersection of Algebraic Cycles". Journal of Mathematical Sciences (Springer New York) 82 (5): 3625–3632. doi:10.1007/bf02362566.
2. ↑ 2.0 2.1 Hatcher, Allen (2001), Algebraic Topology, Cambridge University Press, ISBN 0-521-79540-0
3. ↑ 3.0 3.1 Allen Hatcher. "Chapter 0: Algebraic Topology". http://www.math.cornell.edu/~hatcher/AT/ATch0.pdf. Retrieved August 6, 2010.

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