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