“Genus” has number of distinct but compatible definitions.

In topologyMathworldPlanetmathPlanetmath, if S is an orientable surface, its genus g(S) is the number of “handles” it has. More precisely, from the classification of surfacesMathworldPlanetmath, we know that any orientable surface is a sphere, or the connected sumMathworldPlanetmathPlanetmath of n tori. We say the sphere has genus 0, and that the connected sum of n tori has genus n (alternatively, genus is additive with respect to connected sum, and the genus of a torus is 1). Also, g(S)=1-χ(S)/2 where χ(S) is the Euler characteristicMathworldPlanetmath of S.

In algebraic geometryMathworldPlanetmathPlanetmath, the genus of a smooth projective curve X over a field k is the dimensionMathworldPlanetmath over k of the vector space Ω1(X) of global regularPlanetmathPlanetmathPlanetmath differentials on X. Recall that a smooth complex curve is also a Riemann surface, and hence topologically a surface. In this case, the two definitions of genus coincide.

Title genus
Canonical name Genus
Date of creation 2013-03-22 12:03:45
Last modified on 2013-03-22 12:03:45
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 10
Author mathcam (2727)
Entry type Definition
Classification msc 14H99