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.

If you allow one to be orientable and the other to be non-orientable, then you can find two closed surfaces which are not homeomorphic but have the same Euler characteristic. — Michael Albanese, Mar 13 '18 at 12:45

score 3 · Answer 1 · answered Mar 13 '18 at 11:14

3

Consider a punctured disc or the torus.

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

answered Mar 13 '18 at 11:14

Thomas Rot

10,023

score 1 · Answer 2 · answered Mar 25 '18 at 21:02

1

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.

answered Mar 25 '18 at 21:02

Arnaud Mortier

27,276

score -1 · Answer 3 · answered Mar 13 '18 at 10:01

-1

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

answered Mar 13 '18 at 10:01

Guest

9

1

You are thinking of closed surfaces. Take one closed and one not closed. – YAlexandrov Mar 13 '18 at 10:49
3

It's not "by definition", it's a theorem, the classification of (closed!) surfaces. – Najib Idrissi Mar 13 '18 at 11:20

Question about homeomorphic surfaces

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