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genus of topological surface
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(Definition)
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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 order 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:
Definition 1 The integer of the above theorem is called the genus of the surface.
Theorem 2 Any compact orientable surface without boundary is a connected sum of  tori, where  is its genus.
Remark 1 The previous theorem is the reason why genus is sometimes referred to as “the number of handles”.
Theorem 3 The genus is a complete homeomorphism invariant, i.e. two compact orientable surfaces without boundary are homeomorphic if and only if they have the same genus.
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Cross-references: homeomorphic, homeomorphism, connected sum, property, simple closed curves, cardinality, Betti number, integer, numbers, boundary, manifold, connected, compact, equivalent, topologically equivalent, closed, orientable, invariant, complete, topology, surfaces, topological invariant
There are 9 references to this entry.
This is version 26 of genus of topological surface, born on 2002-08-15, modified 2006-09-11.
Object id is 3296, canonical name is GenusOfTopologicalSurface.
Accessed 8428 times total.
Classification:
| AMS MSC: | 55M99 (Algebraic topology :: Classical topics :: Miscellaneous) |
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Pending Errata and Addenda
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