Euler characteristic of part of the sphere

Question

Let R be the part of the sphere in $R^3$ bounded by two smoothly closed curves that do not intersect.

For instance, R is the region bounded by a great circle and a smaller circle paralle to it.

How to compute the Euler characteristic of R?

What kind of triangualtion can one get? How to visualize that?

Answer 1

A sphere with two disks removed is diffeomorphic to a bounded cylinder and hence has the homotopy type a circle. The Euler characteristic of the circle is then $$\chi(S^1)=\mbox{rk}\,H_0(S^1)-\mbox{rk}\, H_1(S^1)=\mbox{rk}\,\mathbb{Z}-\mbox{rk}\,\mathbb{Z}=1-1=0$$

and so by homotopy invariance of homology (and hence Euler characteristic), we get $$\chi(S^2\setminus(D^2\sqcup D^2))=0.$$

Answer 2

Alternatively: imagine that you have a triangulation of the sphere such that each disk that you remove from it corresponds to one triangle. In terms of this triangulation, the only difference between the sphere and the punctured sphere is that the latter has one less triangle. Therefore $\chi(S^2 - D^2) = \chi(S^2) - 1$. In particular, removing two triangles that share no common face yields $\chi(S^2 - 2D^2) = \chi(S^2) - 2 = 0$.