What surface do I get by attaching $g$ handles as well as $k$ crosscaps to a sphere?
I recently found out that there is a classification of compact connected surfaces that says that every such surface (or, $2$-manifold) is homeomorphic to either $S_g$, the sphere with $g \geq 0$ handles, or $N_k$, the sphere with $k \geq 1$ crosscaps.
If my understanding is correct:
- attaching a handle to a sphere means deleting two discs from the sphere and attaching the ends of a cylinder to the boundaries of these holes. I am able to visualize how attaching one handle to a sphere gives a torus, attaching two handles gives a double torus, etc.
- attaching a crosscap to a sphere means deleting a disc and pasting a Möbius strip to the boundary of this hole.
My question is, what happens if I attach a handle and a crosscap to a sphere? I presume the result is still a compact connected surface (I don't see how it cannot be if every $S_g$ and $N_k$ is). So, by the classification theorem, this surface must also be homeomorphic to some $S_g$ or $N_k$. How can I find out to what surface it will be homeomorphic to? More generally, if I attach $g$ handles and $k$ crosscaps to a sphere, what is the resulting surface homeomorphic to as per the classification theorem?