Simplicial Structures in Topology

184 V Triangulable Manifolds
Fig. V.14 Attaching a handle
(second method)
(second method), to the connected sum with a Klein bottle. The connected sum
with a projective plane, pictured in Fig. V.15, will be one of the fundamental steps
Fig. V.15 Attaching a projective
plane
in the proof of the next theorem, known as the Fundamental Theorem of Closed
Surfaces, which classifies all closed surfaces. It tells us that the only closed surfaces
are exactly the sphere S 2 , nT 2 ,andnRP 2 . Its proof is based on the fact
that any closed surface is triangulable. The existence of a triangulation for closed
surfaces was first proved by Tibor Radó in[29]; we assume this result to be well
known. 2 Here is another fundamental step in the proof: if γ ⊂ S is a (simplicial)
simple closed curve in S, then, there exists a neighbourhood of γ in S, givenby
the union of triangles in a simplicial subdivision of a triangulation of S, which
is homeomorphic to either a cylinder or a Möbius band. By cutting the surface
along γ we obtain, therefore, a new surface with either two (in the case of the cylinder)
or one (in the case of the Möbius band) boundary component. Finally, either
one or two disks D 2 may be attached to these boundaries, in order to obtain
a (triangulated) surface S ′ without boundary. If the surface S ′ is in turn connected,
a homeomorphic copy of S may be constructed by attaching either a handle
(in the case of the cylinder) or a projective plane (in the case of the Möbius
band).
2 See [8] for an elementary proof, based essentially on the Jordan–Schoenflies Theorem: a simple
closed curve J on the Euclidean plain divides it into two regions and there exists a homeomorphism
from the plane in itself that sends J into a circle.