What shape is homeomorphic to a sphere attached with a handle and a cross cap?

Question

The classification theorem for closed surfaces states that a orientable closed surface is homeomorphic to a sphere with m handles with m >= 0; while a nonorintable closed surface is homeomorphic to a sphere with n cross caps with n > 0.

Then, what surface is homeomorphic to a sphere with both a handle and a cross cap? Is this surface orientable? What is its Euler number?

Thanks.

Answer 1

Dyck's theorem shows that although a torus by itself can't be turned into crosscaps, if there is another crosscap around you can send the torus through it and turn it into two crosscaps, so the sphere with one torus and one crosscap is homeomorphic to a sphere with three crosscaps, and more generally, a sphere with $n$ crosscaps and $m$ tori is equivalent to one with $ n+2m $ crosscaps and no tori.