{{Short description|Kind of two-dimensional surface}} thumb|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.<ref>{{cite journal |last1=Brasselet|first1=J. P.|year=1996 |title=Intersection of Algebraic Cycles |journal= Journal of Mathematical Sciences|publisher=Springer New York|volume= 82|issue= 5|pages=3625–3632|doi=10.1007/bf02362566}}</ref>
== Parametrisation ==
A pinched torus is easily parametrisable. Let us write {{nowrap|1=''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 {{nowrap|1=ƒ : [0,2π)<sup>2</sup> → '''R'''<sup>3</sup>}} where: :<math>f(x,y) = \left( g(x,y)\cos x , g(x,y)\sin x , \sin\!\left(\frac{x}{2}\right)\sin y \right) </math>
== Topology ==
Topologically, the pinched torus is homotopy equivalent to the wedge of a sphere and a circle.<ref name="TOP">{{Citation|first=Allen|last=Hatcher|title=Algebraic Topology|publisher=Cambridge University Press|year=2001|isbn=0-521-79540-0}}</ref><ref name="TOP0">{{cite web|url=https://pi.math.cornell.edu/~hatcher/AT/ATch0.pdf|title=Chapter 0: Algebraic Topology|author=Allen Hatcher|accessdate=August 6, 2010}}</ref> It is homeomorphic to a sphere with two distinct points being identified.<ref name="TOP"/><ref name="TOP0"/>
=== Homology ===
Let ''P'' denote the pinched torus. The homology groups of ''P'' over the integers can be calculated. They are given by: :<math>H_0(P,\Z) \cong \Z, \ H_1(P,\Z) \cong \Z, \ \text{and} \ H_2(P,\Z) \cong \Z. </math>
=== Cohomology ===
The cohomology groups of ''P'' over the integers can be calculated. They are given by: :<math>H^0(P,\Z) \cong \Z, \ H^1(P,\Z) \cong \Z, \ \text{and} \ H^2(P,\Z) \cong \Z. </math>
== References == {{reflist}}
Category:Surfaces