Pinched torus

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(\frac{x}{2}\right)\sin y \right)

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,\Z) \cong \Z, \ H_1(P,\Z) \cong \Z, \ \text{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, \ \text{and} \ H^2(P,\Z) \cong \Z.

References

References

1. (1996). "Intersection of Algebraic Cycles". Springer New York.
2. Hatcher, Allen. (2001). "Algebraic Topology". Cambridge University Press.
3. Allen Hatcher. "Chapter 0: Algebraic Topology".