Determine the Euler characteristic of the surface
$$ M=\left\{(x,y,z); \sqrt{x^2+y^2}=1+z^{2n}, 0< z< 1\right\} $$
Determine the Euler characteristic of the surface
$$ M=\left\{(x,y,z); \sqrt{x^2+y^2}=1+z^{2n}, 0< z< 1\right\} $$
For any $n\in \mathbb{R}$, your surface is a cylinder, and homotopic to the circle.
(I don't see why Agusti Roig gets the disjoint union of two cylinders)