What is algebraic topology in layman's terms

Genus (mathematics)

A genus 2 surface

In mathematics , genus (Plural Genera ) has several different but closely related meanings. The most common concept, the genus of a (orientable) surface, is the number of "holes" it has such that a sphere has genus 0 and a torus is genus 1. This is detailed below.


Orientable surfaces

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

The genus of a connected, orientable surface is an integer that represents the maximum number of cuts along non-intersecting closed simple curves, not including the resulting manifold

For example:

  • The ball S.and a disk both have the genus zero.
  • A torus is of genus one, as is the surface of a coffee cup with a handle. This is the source of the joke, "Topologists are people who can't tell their donut from their coffee mug."

An explicit construction of surfaces of genus g can be found in the article on the basic polygon.

  • Type of orientable surfaces
  • Genus 0

  • Genus 1

  • Genus 2

  • Genus 3

In simpler terms, the value of the species of an orientable surface corresponds to the number of "holes" it has.

Non-orientable surfaces

The one that cannot be orientated genus , Demigenus or Euler genus a connected, non-orientable closed surface is a positive integer that represents 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 - k, where k is the non-orientable genus.

For example:


The genus of a node K is defined as the minimal genus of all Seifert surfaces for K. However, a Seifert surface of a node is a manifold with the boundary, where the boundary is the node, ie homeomorphic to the unit circle. The genus of such a surface is defined as the genus of the two-manifold obtained by gluing the unit disk along the boundary.


The genus of a three-dimensional handlebody is an integer representing the maximum number of cuttings along embedded slices

For example:

  • A sphere has the genus zero.
  • A fixed torus D × S has the genus one.

Graph theory

The genus of a graph is the minimal integer n so that the graph can be drawn without crossing on a sphere with n handles (ie an oriented surface of the genus n). Thus, a planar graph is of genus 0 because it can be drawn on a sphere without crossing itself.

The non-orientable genus of a graph is the minimal integer n, so that the graph can be drawn without crossing on a sphere with n cross caps (ie a non-orientable surface of the (non-orientable) genus n). (This number is also called a Demigenus designated.)

The Euler genus is the minimum integer n such that the graph can be drawn without crossing on a sphere with n cross caps or on a sphere with n / 2 handles.

In topological graph theory there are several definitions of the genus of a group. Arthur T. White introduced the following concept. The genus of a group G is the minimal genus of a (connected, undirected) Cayley graph for G.

The problem of the graph genus is NP-complete.

algebraic geometry

There are two related definitions of genus of a projective algebraic scheme X: the arithmetic genus and the geometric genus. If X is an algebraic curve with the complex number definition field and X has no singular points, then these definitions will match and match the topological definition applied to the Riemann surface of X (its complex point manifold). For example, the definition of the elliptic curve from algebraic geometry is related to a non-singular projective curve of genus 1 with a given rational point.

According to the Riemann-Roch theorem, an irreducible plane curve of degree through the vanishing location of a section

where s is the number of singularities if counted correctly.


The genus can also be calculated for the graph that is spanned by the network of chemical interactions in nucleic acids or proteins. In particular, one can study the growth of the genus along the chain. Such a function (referred to as a genus trace) shows the topological complexity and domain structure of biomolecules.

See also