genus of topological surface
The genus is a topological invariant^{} of surfaces^{}. It is one of the oldest known topological invariants and, in fact, much of topology^{} has been created in to generalize this notion to more general situations than the topology of surfaces. Also, it is a complete^{} invariant in the sense that, if two orientable closed surfaces have the same genus, then they must be topologically equivalent. This important topological invariant may be defined in several equivalent^{} ways as given in the result below:
Theorem.
Let $\mathrm{\Sigma}$ be a compact^{}, orientable connected $\mathrm{2}$–dimensional manifold^{} (a.k.a. surface) without boundary. Then the following two numbers are equal (in particular the first number is an integer)

(i)
half the first Betti number of $\mathrm{\Sigma}$
$$\frac{1}{2}dim{H}^{1}(\mathrm{\Sigma};\mathbb{Q})\mathit{\hspace{1em}},$$ 
(ii)
the cardinality of a set $C$ of mutually nonintersecting simple closed curves with the property that $\mathrm{\Sigma}\setminus C$ is a connected surface.
Definition.
The integer of the above theorem^{} is called the genus of the surface.
Theorem.
Any compact orientable surface without boundary is a connected sum^{} of $g$ tori, where $g$ is its genus.
Remark.
The previous theorem is the reason why genus is sometimes referred to as “the number of handles”.
Theorem.
The genus is a homeomorphism , i.e. two compact orientable surfaces without boundary are homeomorphic if and only if they have the same genus.
Title  genus of topological surface 

Canonical name  GenusOfTopologicalSurface 
Date of creation  20130322 12:56:21 
Last modified on  20130322 12:56:21 
Owner  Mathprof (13753) 
Last modified by  Mathprof (13753) 
Numerical id  29 
Author  Mathprof (13753) 
Entry type  Definition 
Classification  msc 55M99 
Synonym  genus 