genus of topological surface

The genus is a topological invariantPlanetmathPlanetmath of surfacesMathworldPlanetmath. It is one of the oldest known topological invariants and, in fact, much of topologyMathworldPlanetmathPlanetmath has been created in to generalize this notion to more general situations than the topology of surfaces. Also, it is a completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 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 equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ways as given in the result below:


Let Σ be a compactPlanetmathPlanetmath, orientable connected 2–dimensional manifoldMathworldPlanetmath (a.k.a. surface) without boundary. Then the following two numbers are equal (in particular the first number is an integer)


The integer of the above theoremMathworldPlanetmath is called the genus of the surface.


Any compact orientable surface without boundary is a connected sumMathworldPlanetmathPlanetmath of g tori, where g is its genus.


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
Canonical name GenusOfTopologicalSurface
Date of creation 2013-03-22 12:56:21
Last modified on 2013-03-22 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