# What is the quantity 2(handles)+crosscaps called?

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 $S_{g,k} \mapsto 2g+k$?

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Isn't this just $\dim H_1$ (say with rational coefficients)? –  Qiaochu Yuan May 5 '11 at 21:09
Thanks. Yes, I thought of using that, but I didn't really want to bring in homology, and I thought there would be a standard name like say generalized genus to use. –  Tony Huynh May 5 '11 at 21:14
I am just wondering how badly I would be bastardizing nomenclature if I just called it genus? –  Tony Huynh May 5 '11 at 21:16
A crosscap is not a surface but a singularity of a generic map of the surface into $\Bbb R^3$. The surface is called the Moebius band. –  Sergey Melikhov May 5 '11 at 21:38
@Tony - The problem with calling this "generalized genus" is that a generalization of a quantity should agree with the original one where they are both defined. However, in this case your generalized genus of a torus is 2, while the genus is 1. This seems not to be ideal. –  Simon Rose May 5 '11 at 22:09

You could certainly call it the Betti number $b_1$ without bringing in homology.

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This is an eample, I believe, where the names started off on a bad track and prevented any better replacement from developing. "Genus" even in the orientable case is often a cause for confusion, and it becomes really terrible with the two kinds of genera. So, people end up using circumlocutions: "the non-orientable surface of Euler charcteristic -2", etc., obscuring their geometrry

If all you want is the function, then $2 - \chi$ is perfectly good.

The trouble is that genus is often used as part of a naming convention, emphasizing the description in terms of semigroup generators for surfaces under connected sum, that is, the torus and the projective plane.

In many cases where these quantities are important, 2-dimensional orbifolds are also important. Surfaces with boundary, and surfaces with points removed, , are also important. The semigroup of 2-dimensional orbifolds is not finitely generated, although generators are easily parametrized. I struggled with getting an appropriate way of talking about these simple objects that had complicated names them for a long time, in particular in discussions with Conway. When I convinced him (talking in terms of all these kinds of circumlocutions) that orbifolds are a good way to think about symmetry groups, he had exactly the right instinct --- he came up with a good systematic notation,. Oriented surfaces are (O^n) for the surface of genus n and $(X^n)$ for the unoriented surface of unoriented genus n. You can read about the rest of the notation elsewhere. Again, the function $2-\chi$ doesn't require another name. Just the surfaces and orbifolds require better names.

Conway has advocated the terminology "double torus" "triple torus" and "n-tuple torus" rather than 2-holed torus etc. which are terms of endless mixup, because people think of punching a holes until they've had much indoctrination.

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Thanks very much for your answer. I agree that n-holed torus is immensely confusing. –  Tony Huynh May 6 '11 at 7:44

I don't know of a name for the specific quantity $2g + k$, but it should be noted that $2 - (2g+k) = \chi(S_{g,k})$, the Euler characteristic.

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Indeed. This is another reason why this is the more natural definition, among others. –  Tony Huynh May 5 '11 at 21:02
Indeed. So why not just call it $2-\chi$? –  Mark Grant May 5 '11 at 21:48
@Mark - Yeah, that's sort of what I was getting at. –  Simon Rose May 5 '11 at 22:07
I am using the notation $2- \chi$ for it, but am just wondering if it has a widely accepted name. –  Tony Huynh May 5 '11 at 22:14
$2 - \chi$ is a widely accepted name. –  Ryan Budney May 5 '11 at 22:40
My colleague, Chris Cooper, has some unpublished notes in which he calls this quantity the $\it weight$ of the surface. I'm not aware of anyone else using this terminology.