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The green surface pictured above has 2 holes of the relevant sort. For instance: The sphere and a disc both have genus zero. A torus has genus one, as does the surface of a coffee mug with a handle. This is the source of the joke "topologists are people who can't tell their donut from their coffee mug."
As examples, a genus zero surface (without boundary) is the two-sphere while a genus one surface (without boundary) is the ordinary torus. The surfaces of higher genus are sometimes called n-holed tori (or, rarely, n-fold tori). The terms double torus and triple torus are also occasionally used.
In the case of genus one, a fundamental convex polygon is sought for the action by translation of Λ = Z a ⊕ Z b on R 2 = C where a and b are linearly independent over R. (After performing a real linear transformation on R 2, it can be assumed if necessary that Λ = Z 2 = Z + Z i; for a genus one Riemann surface it can be taken to have the form Λ = Z 2 = Z + Z ω, with Im ω > 0.)
It is convenient to combine the two families by regarding the sphere as the connected sum of 0 tori. The number g of tori involved is called the genus of the surface. The sphere and the torus have Euler characteristics 2 and 0, respectively, and in general the Euler characteristic of the connected sum of g tori is 2 − 2g.
The term double torus is occasionally used to denote a genus 2 surface. [4] [5] A non-orientable surface of genus two is the Klein bottle. The Bolza surface is the most symmetric Riemann surface of genus 2, in the sense that it has the largest possible conformal automorphism group. [6] Representations of genus 2 surfaces
All have a standard splitting of genus one. This is the image of the Clifford torus in S 3 {\displaystyle S^{3}} under the quotient map used to define the lens space in question. It follows from the structure of the mapping class group of the two-torus that only lens spaces have splittings of genus one.
A g-holed toroid can be seen as approximating the surface of a torus having a topological genus, g, of 1 or greater. The Euler characteristic χ of a g holed toroid is 2(1-g). [2] The torus is an example of a toroid, which is the surface of a doughnut. Doughnuts are an example of a solid torus created by rotating a disk, and are not toroids.
A torus. The next case is a Riemann surface of genus =, such as a torus /, where is a two-dimensional lattice (a group isomorphic to ). Its genus is one: its first singular homology group is freely generated by two loops, as shown in the illustration at the right.