Simplicial Structures in Topology

188 V Triangulable Manifolds
a tree. When χ(KT )=1, V is then homeomorphic to the disk D 2 and S is obtained
by gluing two disks on the boundaries; it follows that S ∼ = S 2 . Instead, if χ(KT ) < 1,
then the homology H1(KT ) is non-trivial and so there is at least one non-trivial
simple cycle, denoted by γ, in the graph KT . Let us consider the simplicial complex
Kγ obtained by cutting K along γ (in other words, we duplicate all vertices and edges
of γ). The resulting triangulated surface Sγ = |Kγ| is still connected: indeed, the tree
T and the cycle γ are disjoint, but T has a vertex in the interior of each triangle of
K, and therefore also of Kγ. It is a surface with boundary: if γ has a neighbourhood
homeomorphic to a Möbius band, then its boundary has one component; on the
other hand, if the neighbourhood of γ is a cylinder, there are two components. Let
us consider the cones on the components of the boundary of Kγ and attach them to
the holes that we have created: we end up with a new triangulated surface S ′ ∼ = |K ′ |.
The tree T is easily extended to a tree T ′ ⊂ K ′ by adding a vertex to the barycentre of
every triangle of the attached cones and an edge crossing the edge of the boundary
of the attachment, as in Fig. V.17.Letl be the number of edges of the component of
Fig. V.17 Extension of T to
the cone on the boundary
component
such a boundary. As it is easily inferred from Fig. V.17, by attaching a triangulated
disk (cone on the component of γ), l sides are removed from the graph KT , whereas
one vertex (the centre of the cone) and l edges through such a vertex are added.
The Euler–Poincaré characteristic of the corresponding graph K ′ T ′, obtained in this
manner, is
χ(K ′ T ′)=
�
χ(KT )+1 if Kγ has one boundary component
χ(KT )+2 if Kγ has two boundary components.
We note that the process of cutting along γ may be reversed: |K| is obtained by
attaching a handle (in the case of two components) to |K ′ | or a projective plane (in
the case of a single component). By repeating this procedure, we must end up with
nothing other than a sphere. 4
We have therefore proved that every surface is the connected sum of a sphere,
a certain number nT of tori, a certain number nK of Klein bottles, and a certain
number nP of projective planes. However, since the Klein bottle is homeomorphic
4 For a proof based on identifying polygons, see William Massey [25, Theorem 1.5.1].