Earl Richard, 'Making surfaces',
Topology: A Very Short Introduction (Oxford, 2019)
page 39
Classification is an important theme in mathematics. A mathematical theory often begins with definitions and rules about certain mathematical objects or structures (say functions or curves) and seeks to prove results about them using those rules. It’s natural to search for examples satisfying those rules, preferably producing a complete list or classification of such objects.
We are now close to classifying closed surfaces. Explicitly, we are seeking to give a complete list of all the closed surfaces, so that every closed surface is homeomorphic to (i.e. topologically the same as) one of the surfaces on the list, and the list contains no duplicates—each surface on the list can be shown to be topologically different from all others on the list.
It turns out that the Euler number goes a long way to separating out the different surfaces, but we have seen that this cannot be the whole story as the torus and Klein bottle have the same Euler number whilst being different surfaces—the first is orientable, the second not. The only missing ingredient in the classification is that notion of orientability.
So the first half of the classification theorem for two-sided surfaces states:
- An orientable closed surface is homeomorphic to precisely one of the tori $\mathbb T^{\#g}$ where $g=0,1,2\ldots$ These tori are not topologically the same as one another as they have different Euler numbers—the Euler number of $T^{\#g}$ is $2-2g$.
A similar result holds for one-sided closed surfaces. Just as the torus $\mathbb T$ is a building block for the orientable surfaces, so can the projective plane $ℙ$ be used to make the non-orientable surfaces. Recall that the projective plane $ℙ$ has Euler number 1. So the connected sums $ℙ\#ℙ$ and $ℙ\#ℙ\#ℙ$ have
Euler number of $ℙ\#ℙ= 1 + 1 - 2 = 0$,
Euler number of $ℙ\#ℙ\#ℙ= 0 + 1 - 2 = –1$,
and more generally $k$ copies of $ℙ$ in a connected sum, a surface denoted $ℙ^{\#k}$, has Euler number $2-k$.
And the second half of the classification theorem for one-sided surfaces states:
- A non-orientable closed surface is homeomorphic to precisely one of $ℙ^{\#k}$ where $g=1,2,3\ldots$
These surfaces are not topologically the same as they have different Euler numbers—the Euler number of $ℙ^{\#k}$ is $2-k$.
Making a connected sum with $ℙ$ is equivalent to sewing a Möbius strip into the surface. $ℙ$ itself can be made by introducing a Möbius strip into a sphere; to do this we might make a tear in the sphere and then, rather than gluing the tear back together, we could instead assign reverse arrows to the two sides of the tear, thus introducing a Möbius strip. So the surface $ℙ^{\#k}$ can be thought of as a sphere with $k$ Möbius strips sewed in.
Overall then, the classification theorem says that if we know the Euler number of a closed surface and whether it is one- or two-sided, then we know its topological shape. If you were wondering, where the Klein bottle is on this list, we know its Euler number to be 0 and we know it to be one-sided. The only surface in the classification matching these facts is the $k=2$ surface $ℙ\#ℙ$ and this is topologically the same as the Klein bottle. We might create a yet more complicated connected sum such as $\mathbb T\#\mathbb T\#ℙ\#ℙ\#ℙ$ which at first glance is not on our list. This surface is one-sided and its Euler number equals
(Euler number of $\mathbb T\#\mathbb T$) + (Euler number of $ℙ\#ℙ\#ℙ$) – 2= (2 – 2 × 2) + (2 – 3) – 2 = –5,
so topologically it’s the same surface as $ℙ^{\#7}$. And at long last we are able to identify the surface we formed in Figure 14. That surface had Euler number $–1$ and so the surface is $ℙ\#ℙ\#ℙ$, this being the only surface on our list with that Euler number. | 
