A genus-2 surface

In mathematics, genus (pl.: genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface.^[1] A sphere has genus 0, while a torus has genus 1.

Topology

Orientable surfaces

The coffee cup and donut shown in this animation both have genus one.

The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected.^[2] It is equal to the number of handles on it. Alternatively, it can be defined in terms of the Euler characteristic χ, via the relationship χ = 2 − 2g for closed surfaces, where g is the genus. For surfaces with b boundary components, the equation reads χ = 2 − 2g − b. In layman's terms, it's the number of "holes" an object has ("holes" interpreted in the sense of doughnut holes; a hollow sphere would be considered as having zero holes in this sense). A torus has 1 such hole, while a sphere has 0. The green surface pictured above has 2 holes of the relevant sort.

For instance:

• The sphere S² 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."

Explicit construction of surfaces of the genus g is given in the article on the fundamental polygon.

• Genus of orientable surfaces
• 
  Planar graph: genus 0
• 
  Toroidal graph: genus 1
• 
  Teapot: Double Toroidal graph: genus 2
• 
  Pretzel graph: genus 3

In simpler terms, the value of an orientable surface's genus is equal to the number of "holes" it has.^[3]

Algebraic geometry

There are two related definitions of genus of any projective algebraic scheme X: the arithmetic genus and the geometric genus.^[7] When X is an algebraic curve with field of definition the complex numbers, and if X has no singular points, then these definitions agree and coincide with the topological definition applied to the Riemann surface of X (its manifold of complex points). For example, the definition of elliptic curve from algebraic geometry is connected non-singular projective curve of genus 1 with a given rational point on it.

By the Riemann–Roch theorem, an irreducible plane curve of degree ${\displaystyle d}$ given by the vanishing locus of a section ${\displaystyle s\in \Gamma (\mathbb {P} ^{2},{\mathcal {O}}_{\mathbb {P} ^{2}}(d))}$ has geometric genus

${\displaystyle g={\frac {(d-1)(d-2)}{2}}-s,}$

where ${\displaystyle s}$ is the number of singularities when properly counted.

Differential geometry

In differential geometry, a genus of an oriented manifold ${\displaystyle M}$ may be defined as a complex number ${\displaystyle \Phi (M)}$ subject to the conditions

• ${\displaystyle \Phi (M_{1}\amalg M_{2})=\Phi (M_{1})+\Phi (M_{2})}$
• ${\displaystyle \Phi (M_{1}\times M_{2})=\Phi (M_{1})\cdot \Phi (M_{2})}$
• ${\displaystyle \Phi (M_{1})=\Phi (M_{2})}$ if ${\displaystyle M_{1}}$ and ${\displaystyle M_{2}}$ are cobordant.

In other words, ${\displaystyle \Phi }$ is a ring homomorphism ${\displaystyle R\to \mathbb {C} }$, where ${\displaystyle R}$ is Thom's oriented cobordism ring.^[8]

The genus ${\displaystyle \Phi }$ is multiplicative for all bundles on spinor manifolds with a connected compact structure if ${\displaystyle \log _{\Phi }}$ is an elliptic integral such as ${\displaystyle \log _{\Phi }(x)=\int _{0}^{x}(1-2\delta t^{2}+\varepsilon t^{4})^{-1/2}dt}$ for some ${\displaystyle \delta ,\varepsilon \in \mathbb {C} .}$ This genus is called an elliptic genus.

The Euler characteristic ${\displaystyle \chi (M)}$ is not a genus in this sense since it is not invariant concerning cobordisms.

Biology

Genus can be also calculated for the graph spanned by the net of chemical interactions in nucleic acids or proteins. In particular, one may study the growth of the genus along the chain. Such a function (called the genus trace) shows the topological complexity and domain structure of biomolecules.^[9]

See also

• Group (mathematics)
• Arithmetic genus
• Geometric genus
• Genus of a multiplicative sequence
• Genus of a quadratic form
• Spinor genus

Citations

References