In mathematics, a genus ''g'' surface (also known as a ''g''-torus or ''g''-holed torus) is a

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 ...

formed by the

connected sum In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ...

of ''g'' many tori: the interior of a disk is removed from each of ''g'' many tori and the boundaries of the ''g'' many disks are identified (glued together), forming a ''g''-torus. The

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...

of such a surface is ''g''. A genus ''g'' surface is a

two-dimensional In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...

manifold. The classification theorem for surfaces states that every

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 ...

connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...

two-dimensional manifold is homeomorphic to either the sphere, the connected sum of tori, or the connected sum of

real projective plane In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...

Definition of genus

The genus of a connected orientable surface is an

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 languag ...

representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It is equal to the number of

handles A handle is a part of, or attachment to, an object that allows it to be grasped and manipulated by hand. The design of each type of handle involves substantial ergonomic issues, even where these are dealt with intuitively or by following t ...

on it. Alternatively, it can be defined in terms of the Euler characteristic ''χ'', via the relationship ''χ'' = 2 − 2''g'' for closed surfaces, where ''g'' is the genus. The genus (sometimes called the demigenus or Euler genus) of a connected non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere. Alternatively, it can be defined for a closed surface in terms of the Euler characteristic ''χ'', via the relationship ''χ'' = 2 − ''g'', where ''g'' is the non-orientable genus.

Genus 0

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 i ...

surface of genus zero is the

sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...

''S''^''2''. Another surface of genus zero is the disc. File:Sphere filled blue.svg, A sphere

S^2

File:1-ball.svg, A closed disc (with boundary)

Genus 1

A genus one orientable surface is the ordinary torus. A non-orientable surface of genus one is the

projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...

Elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...

s over the complex numbers can be identified with genus 1 surfaces. The formulation of elliptic curves as the embedding of 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 tou ...

in the

complex projective plane In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates :(Z_1,Z_2,Z_3) \in \mathbf^3,\qquad (Z_1, ...

follows naturally from a property of

Weierstrass's elliptic functions In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by t ...

that allows elliptic curves to be obtained from the quotient of the complex plane by a

lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...

. File:Torus illustration.png, A torus of genus 1 File:Elliptic curve simple.png, An elliptic curve

Genus 2

The term double torus is occasionally used to denote a genus 2 surface. A non-orientable surface of genus two is the Klein bottle. The

Bolza surface In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (introduced by ), is a compact Riemann surface of genus 2 with the highest possible order of the conformal automorphism group in this genus, namely GL_2(3) of order 48 ...

is the most symmetric

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...

2, in the sense that it has the largest possible conformal

automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...

. File:Double torus illustration.png, A torus of genus 2

Genus 3

The term triple torus is also occasionally used to denote a genus 3 surface. The

Klein quartic In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus with the highest possible order automorphism group for this genus, namely order orientation-preserving automorphisms, and automorphisms ...

is a compact

with the highest possible order

for compact Riemann surfaces of genus 3. It has namely order orientation-preserving automorphisms, and automorphisms if orientation may be reversed. File:Sphere with three handles.png, A sphere with three

File:Triple torus array.png, The

of three tori File:Triple torus illustration.png, Triple torus File:Dodecagon with opposite faces identified.svg,

Dodecagon In geometry, a dodecagon or 12-gon is any twelve-sided polygon. Regular dodecagon A regular dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry and rotational sym ...

with opposite edges identifiedJürgen Jost, (1997) "Compact Riemann Surfaces: An Introduction to Contemporary Mathematics", Springer File:14-gon with opposite faces identified.svg, Tetradecagon with opposite edges identified

References

Sources

* James R. Munkres, ''Topology, Second Edition'', Prentice-Hall, 2000, {{ISBN, 0-13-181629-2. * William S. Massey, ''Algebraic Topology: An Introduction'', Harbrace, 1967. Topology Geometry Surfaces