Earl Richard, 'Making surfaces',
Topology: A Very Short Introduction (Oxford, 2019)
Given two closed surfaces S1 and S2, then we can create their connected sum S1#S2. This is a way to glue surfaces together and a useful means of making new surfaces from the few we have so far met. Say S1 and S2 each has a subdivision that includes a ‘triangular’ face bounded by three edges. (The inverted commas here hint that, this being topology, the faces may not be that recognizably triangular in terms of having straight edges.) The connected sum S1#S2 is then created by removing these two triangular faces, so making two holes in the surfaces, and then gluing the two surfaces together along the boundaries of the holes, pairing up the three vertices and three edges with those on the boundary of the second removed face as, for example, in Figure 16.
Figure 16.
Connected sums with tori (a) One torus with a “triangle” missing, (b) A torus with two holes as a connected sum.
Helpfully there is a formula for the Euler number of S1#S2. In making the connected sum, we remove two triangular faces, the six different vertices on these triangles are glued to make three vertices on the connected sum, and likewise six edges are glued to make three. So the total number of faces has gone down by 2 and the total numbers of edges and vertices have each gone down by 3. As V and F are added in the formula for the Euler number, and E is subtracted, overall we have
Euler number of S1#S2 = (Euler number of S1) + (Euler number of S2) – 2.
Or, if you prefer a more careful algebraic proof, say the original subdivision of S1 has V1 vertices, E1 edges, and F1 faces and define V2, E2, F2 similarly for S2. The number of vertices V#, edges E#, and faces F# on the connected sum is given by
V# = V1 + V2 – 3 , E# = E1 + E2 – 3 , F# = F1 + F2 – 2.
Finally
\begin{array}l\text{Euler number of }S_1\#S_2\text{ }=\text{ }V_\#-E_\#+F_\#\\\text{ }=\text{ }{(V_1+V_2-\text{ }3) }-\text{ }{(E_1+E_2-\text{ }3) }+\text{ }{(F_1+F_2-\text{ }2)}\\\text{ }=\text{ }{(V_1-E_1+F_1) }+\text{ }{(V_2-E_2+F_2) }+\text{ }{(-3\text{ }+\text{ }3\text{ }-\text{ }2)}\\\text{ }={(\text{Euler number of }S_1) }+\text{ }{(\text{Euler number of }S_2) }-\text{ }2.\end{array}
Thinking in terms of connected sums helps us work out the Euler numbers of some more complicated surfaces. We know that a torus 𝕋 has an Euler number of 0. The connected sum 𝕋#𝕋 is a torus with two holes (Figure 16(b)) and we see
Euler number of 𝕋#𝕋 = 0 + 0 − 2 = −2
and similarly the torus with three holes, 𝕋#𝕋#𝕋 , has Euler number
(Euler number of 𝕋#𝕋) + (Euler number of 𝕋) − 2 = −2 + 0 − 2 = −4.
In fact, we can see that every time we make a connected sum with 𝕋 the surface gains one more hole and the Euler number reduces by 2. So the torus with g holes—which can be considered as 𝕋#g, the connected sum of g copies of the torus 𝕋 —has Euler number
\[\text{Euler number of }\mathbb T^{\#g} = 2 - 2g.\]
The number g of holes in the surface 𝕋#g is called the genus of the surface. |
