Homology of compact non-orientable surfaces
From Topospaces
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 non-orientable surface
Get more specific information about compact non-orientable surface | Get more computations of homology
Contents
Statement
Suppose is a positive integer. We denote by (not standard notation, should try to find something) the connected sum of the real projective plane with itself times, i.e., the connected sum of copies of the real projective plane.
Unreduced version over the integers
We have:
Reduced version over the integers
We have:
Unreduced version over a module
If we consider the homology with coefficients in a module over a ring where 2 is invertible, then we have:
Homology groups with integer coefficients in tabular form
We illustrate how the homology groups work for small values of . Note that is zero and all higher are zero.
1 | real projective plane | 0 | ||
2 | Klein bottle | 0 | ||
3 | Dyck's surface | 0 |
Related invariants
These are all invariants that can be computed in terms of the homology groups.
Invariant | General description | Description of value for connected sum of copies of real projective plane | 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 . Note that if the Euler characteristic of a compact surface is odd and at most , then the surface must be non-orientable and its homeomorphism type can be computed (using . If the Euler characteristic is even and at most , then there is a unique possibility for a compact orientable surface and a unique possibility for a compact non-orientable surface. For an Euler characteristic of , there is a unique compact orientable surface and no compact non-orientable surface. For an Euler characteristic bigger than , there is no (connected) compact surface possible. |