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 $S\_{g,k} \mapsto 2g+k$?



What is the quantity 2(handles)+crosscaps called?
You could certainly call it the [Betti number](http://en.wikipedia.org/wiki/Betti_number) $b\_1$ without bringing in homology. 


