It is well-known that up to homeomorphism, the complete set of orientable surfaces is $\lbrace S_g : g=0,1,\dots \rbrace$, where $S_g$ is the sphere with $g$ handles. The complete set of non-orientable surfaces is $\lbrace N_k : k=1,2, \dots \rbrace$, where $N_k$ is the sphere with $k$ crosscaps.

Typically the *genus* of $S_g$ is defined to by $g$, and the *genus* or sometimes *non-orientable genus* of $N_k$ is defined to be $k$. I would actually prefer to define the *genus* of $S_g$ to be $2g$ and the genus of $N_k$ to be $k$. In some sense this is more natural since if $S_{g,k}$ is a sphere with $g$ handles and $k\geq 1$ crosscaps, then $S_{g,k} \cong N_{2g+k}$. Moreover, I am writing a proof where I want to proceed by induction on some sort of genus. It seems more natural that my invariant should go down by 1 when moving from $S_{g,1}$ to $S_{g,0}$ instead of going down by $g+1$.

**Question.** What is the name for the invariant $S_{g,k} \mapsto 2g+k$?

namelike saygeneralized genusto use. – Tony Huynh May 5 '11 at 21:14genus? – Tony Huynh May 5 '11 at 21:16