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:
half the first Betti number of
The integer of the above theorem is called the genus of the surface.
The previous theorem is the reason why genus is sometimes referred to as “the number of handles”.
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|
|Date of creation||2013-03-22 12:56:21|
|Last modified on||2013-03-22 12:56:21|
|Last modified by||Mathprof (13753)|