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

genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...

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

two-dimensional A two-dimensional space is a mathematical space with two dimensions, meaning points have two degrees of freedom: their locations can be locally described with two coordinates or they can move in two independent directions. Common two-dimension ...

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

. 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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

connected two-dimensional manifold is

homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...

to either the sphere, the connected sum of tori, or the connected sum of

real projective plane In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...

Definition of genus

The genus of a connected orientable surface is an

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant

disconnected. It is equal to the number of handles on it. Alternatively, it can be defined in terms of the

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...

''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

An orientable surface of genus zero is the

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

''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) File:Minimal_4_colour_sphere.svg, By the Heawood conjecture, it can be coloured with up to 4 mutually adjacent regions

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 (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...

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 the ...

s over the

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s can be identified with genus 1 surfaces. The formulation of elliptic curves as the embedding of a

torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...

in the

complex projective plane In mathematics, the complex projective plane, usually denoted or 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 \C^3, \qquad (Z_1,Z_2, ...

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 is also referred to as ℘-functions and they are usually denoted by the s ...

that allows elliptic curves to be obtained from the quotient of the

complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...

by a lattice. File:Torus illustration.png, A torus of genus 1 File:7_colour_torus.svg, It can be coloured with up to 7 mutually adjacent regions File:Elliptic curve simple.svg, 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 In mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the ...

. The Bolza surface 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 vers ...

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 File:Tietze_genus_2_colouring.svg, It can be coloured with up to 8 mutually adjacent regions

Genus 3

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

with the highest possible order

for compact Riemann surfaces of genus 3. It has orientation-preserving automorphisms, and automorphisms altogether. File:Sphere with three handles.png, A sphere with three handles File:Triple torus array.png, The

of three tori File:Triple torus illustration.png, Triple torus File:Taxel_genus_3_colouring.svg, It can be coloured with up to 9 mutually adjacent regions File:Dodecagon with opposite faces identified.svg,

Dodecagon In geometry, a dodecagon, or 12-gon, is any twelve-sided polygon. Regular dodecagon A regular polygon, 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 ...

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