In mathematics, and especially

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

and differential geometry, a pinched torus (or croissant surface) is a kind of two-dimensional

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...

. It gets its name from its resemblance to a

torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not ...

that has been pinched at a single point. A pinched torus is an example of an

orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space ...

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

2-dimensional

pseudomanifold In mathematics, a pseudomanifold is a special type of topological space. It looks like a manifold at most of its points, but it may contain singularities. For example, the cone of solutions of z^2=x^2+y^2 forms a pseudomanifold. A pseudomanifol ...

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\right)\sin y \right)

Topology

Topologically, the pinched torus is

homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...

equivalent to the

wedge A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converti ...

of a sphere and a circle. It is homeomorphic to a sphere with two distinct points being

identified ''Identified'' is the second studio album by Vanessa Hudgens, released on July 1, 2008 in the U.S. June 24, 2008 in Japan, February 13, 2009 in most European countries and February 16, 2009 in the United Kingdom. The album r ...

Homology

Let ''P'' denote the pinched torus. The

homology group In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topol ...

s of ''P'' over the

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s can be calculated. They are given by: :

H_0(P,\Z) \cong \Z, \ H_1(P,\Z) \cong \Z, \ \text \ H_2(P,\Z) \cong \Z.

Cohomology

The

cohomology group In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...

s of ''P'' over the

s can be calculated. They are given by: :

H^0(P,\Z) \cong \Z, \ H^1(P,\Z) \cong \Z, \ \text \ H^2(P,\Z) \cong \Z.

References

{{reflist Surfaces