Euler characteristic of a surface

Question

I'm suppose to find the Euler characteristic of the surface $$M = \{(x,y,z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^6 =1. \}$$

I know I have to triangulate the surface and $\chi(M) = V-E+F$ with V=vertices, E=edges and F=faces, of the triangles, but I'm really lost in the idea of triangulating this, do I have to give a parametrization like $\big(x,y,(1-x^2-y^2)^{1/6}\big) \mapsto (x,y)$ or something like this? I'm really lost, I need a lot of help please.

Answer 1

Hint:

In general, your best bet for these is to find a homeomorphism to something which you do know the Euler characteristic of. In this case, this equation looks awfully similar to that of the sphere!

What should such a homeomorphism be? How do homeomorphisms effect the Euler characteristic?

Answer 2

Probably this will not help you, but here is an approach along the lines of Morse theory. Slicing with planes perpendicular to the $ z $-axis, each slice is a circle, except for one point at the top $ z = 1 $ and one point at the bottom with $ z = - 1 $. Euler characteristic of circle is 0 (one vertex, one edge), and Euler characteristic of point is 1 (one vertex, no edges). Only the two points contribute to give $ \chi = 2 $.