The Euler characteristic
The Euler characteristic V − E + F on the sphere, the torus and surfaces with handles, gluing a polygon by a word, orientability and the classification of surfaces.
How to solve it
Draw a grid on a surface and the number $V - E + F$ depends not on the grid but only on the surface. That is the Euler characteristic: $2$ for the sphere, $0$ for the torus, $2 - 2g$ for a surface with $g$ handles. Together with orientability it determines a closed surface completely.
Step by step
- A grid: compute $\chi = V - E + F$.
- An orientable surface with $g$ handles: $\chi = 2 - 2g$, so $g = \frac{2 - \chi}{2}$.
- Gluing a polygon by a word: if some letter appears twice with the same exponent ($a \ldots a$), that gluing has a twist — the surface is non-orientable.
- A non-orientable surface is a sum of $k$ projective planes: $\chi = 2 - k$. A connected sum lowers the characteristic: $\chi(A \# B) = \chi(A) + \chi(B) - 2$.
Common mistakes
- Mixing up the signs: edges are subtracted, faces added.
- Calling the gluing $a\,b^{-1}\,a^{-1}\,b^{-1}$ orientable without noticing that $b$ appears twice with the same sign.
- Using $2 - 2g$ for a non-orientable surface instead of $2 - k$.
- Adding the characteristics in a connected sum and forgetting to subtract $2$ — a disc from each part.