Homology of compact orientable surfaces
This article describes the value (and the process used to compute it) of some homotopy invariant(s) for a topological space or family of topological spaces. The invariant is homology and the topological space/family is compact orientable surface
Get more specific information about compact orientable surface | Get more computations of homology
Suppose is a nonnegative integer. We denote by the compact orientable surface of genus , i.e., is the connected sum of copies of the 2-torus (and it is the 2-sphere when ). In other words, , , , and so on.
Unreduced version over the integers
In other words, the zeroth and second homology groups are both free of rank one, and the first homology group is , i.e., the free abelian group of rank .
Reduced version over the integers
Unreduced version with coefficients in
With coefficients in a module over a ring , we have:
Reduced version with coefficients in
These are all invariants that can be computed in terms of the homology groups.
|Invariant||General description||Description of value for genus surface||Comment|
|Betti numbers||The Betti number is the rank of the torsion-free part of the homology group.||, , all higher are zero|
|Poincare polynomial||Generating polynomial for Betti numbers|
|Euler characteristic||In particular, this means that the Euler characteristic is negative for , and also that the Euler characteristic and genus can be computed in terms of each other -- given the genus, we can find the Euler characteristic, and vice versa.|