Examples
The Euler characteristic can be calculated easily for general surfaces by finding a polygonization of the surface (that is, a description as a CW-complex) and using the above definitions.
| Name | Image | Euler characteristic |
|---|---|---|
| Interval | 1 | |
| Circle | 0 | |
| Disk | 1 | |
| Sphere | 2 | |
| Torus (Product of two circles) |
0 | |
| Double torus | −2 | |
| Triple torus | −4 | |
| Real projective plane | 1 | |
| Möbius strip | 0 | |
| Klein bottle | 0 | |
| Two spheres (not connected) (Disjoint union of two spheres) |
2 + 2 = 4 | |
| Three spheres (not connected) (Disjoint union of three spheres) |
2 + 2 + 2 = 6 |
Any contractible space (that is, one homotopy equivalent to a point) has trivial homology, meaning that the 0th Betti number is 1 and the others 0. Therefore its Euler characteristic is 1. This case includes Euclidean space of any dimension, as well as the solid unit ball in any Euclidean space — the one-dimensional interval, the two-dimensional disk, the three-dimensional ball, etc.
The n-dimensional sphere has Betti number 1 in dimensions 0 and n, and all other Betti numbers 0. Hence its Euler characteristic is — that is, either 0 or 2.
The n-dimensional real projective space is the quotient of the n-sphere by the antipodal map. It follows that its Euler characteristic is exactly half that of the corresponding sphere — either 0 or 1.
The n-dimensional torus is the product space of n circles. Its Euler characteristic is 0, by the product property.
Read more about this topic: Euler Characteristic
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