# Euler characteristic of a compact surface

Determine the Euler characteristic of the surface

$$M=\left\{(x,y,z); \sqrt{x^2+y^2}=1+z^{2n}, 0< z< 1\right\}$$

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For $n \in \mathbb{N}$ (including n=0), you just have one cylinder. For non-positive integers, the disjoint union of two cylinders. –  a.r. Aug 26 '10 at 13:00
Sounds like homework... –  Aryabhata Aug 26 '10 at 14:49
This doesn't look compact to me... –  Dylan Wilson Aug 26 '10 at 14:59

For any $n\in \mathbb{R}$, your surface is a cylinder, and homotopic to the circle.