Question about homeomorphic surfaces

Question

I am wondering if there exist an example of two surfaces such that they are orientable (or not orientable) and have same Euler characteristic but they are NOT homeomorphic.

Thanks for your support.

Answer 1

No. By definition, both these surfaces would be homeomorphic to the same connect sum of tori, and so homeomorphic.

Answer 2

Consider a punctured disc or the torus.

This cannot happen for closed surfaces by the classification theorem of surfaces.

Answer 3

The Möbius band and the usual annulus have the same Euler characteristic, since they both deformation retract on a circle.

However, one of them is orientable and the other is not, so they are not homeomorphic.