genus
“Genus” has number of distinct but compatible definitions.
In topology, if S is an orientable surface, its genus g(S) is the number of “handles” it has.
More precisely, from the classification of surfaces
, we know that any orientable
surface is a sphere, or the connected sum
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 characteristic
of S.
In algebraic geometry, the genus of a smooth projective curve X over a field k is the
dimension
over k of the vector space Ω1(X) of global regular
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 |