Non-Integer Genus?

TL;DR Can the notion of genus be extended from a (non-negative) integer concept? Eg to $\mathbb Z$, $\tfrac12 \mathbb Z$ or even $\mathbb R$?

Various things in maths are naturally integers, but are extended to general real numbers. Examples that spring to mind include the following.

• Lattices: eg $\mathbb Z^d$, may be defined for non-integer $d$
• Differentiability: a function can have an $\alpha$-derivative for any $\alpha \in \mathbb R$ (this is something to do with Sobolev norms)
• Powers: defining $x^2 = x \cdot x$ is natural; defining $x^{1/2}$ to be so that $x^{1/2} \cdot x^{1/2}$ is natural enough; defining $x^\alpha$ for $\alpha \in \mathbb R$ is a little less natural

How about the genus of a surface? (This seems most related to a surface's having non-integer dimension.)

My primary concern Euler's polyhedral formula: $V + F - E = 2 - 2g$, where $V$ is the number of vertices, $F$ the number of faces, $E$ the number of edges and $g$ the genus of a polyhedral. In principle one could allow $g \in \tfrac12 \mathbb Z$ to be a half-integer, but I can't think how one would define vertices/faces/edges in a non-integer way!

Having negative genus doesn't seem like an issue, though.

Solution 1:

Euler's polyhedral formula: $V + F - E = 2 - 2g$ also holds for Riemann Surfaces. However, if you try to calculate the Euler characteristic of a closed unit disc in the complex plane: $D := [{z \in \mathbb{C} : \mid{z} \mid \le 1}]$. We can triangulate this, loosely speaking, split up into sections with are of the form of 3 edges and 3 vertices. We can deform the closed disc into shapes which are (topologically speaking) triangles. We can show the closed unit disc as 2 triangles, which we have 4 vertices, 5 edges (some edges are shared), 2 faces. So from this we get $4 + 2 - 5 = 1$. We define the genus to be $g(s) = 1 - \frac{\chi(S)}{2}$, where $\chi(s) = V + F - E$. So, we get $g(s) = \frac{1}{2}$. This shows that this is not a Riemann Surface, but we know the closed disc is already not a Riemann Surface as its not Hausdorff. This is just an example from the complex plane.