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Proof.
See [Radó25]. Incidentally, with regard to the first statement in the theorem,
if the manifold is compact, then the triangulation will be finite. The infinite triangu-
lations occur with noncompact surfaces such as
R
2
, which is not a surface using the
definition above.
In light of Theorem 6.5.1, we shall drop the adjective “combinatorial” in this
chapter and simply refer to a “surface.”
The reader may wonder why we have bothered with a rather technical definition
of a surface, when we have the more natural manifold definition that corresponds to
one's usual intuitive notion about the kind of space that a surface really is. The reason
is that our goal is to give a complete classification of surfaces and for this it is con-
venient to work with proper triangulations. Thus, the technical aspects would not have
been avoided. At least with our choice of definition we do not have to appeal to a
theorem whose proof would involve a lengthy digression if we were to give it.
Note 2.
The second part of Theorem 6.5.1 is important because a space can be tri-
angulated in many ways. It would be a very unsatisfactory state of affairs if some tri-
angulations were proper and others not.
Note 3.
Radó's proof that every two-dimensional topological manifold can be trian-
gulated used methods from complex analysis. The obvious generalization that every
topological n-dimensional manifold can be triangulated remained a famous unsolved
problem called the
triangulation problem
. We need to point out though that histori-
cally when searching for a triangulation for a manifold one was not satisfied with just
any triangulation. One assumed a weak regularity condition on the “star” of each
vertex.
Definition.
Let K be a simplicial complex and s a simplex in K. The
star
of s,
denoted by star(s), is the union of all the simplices of K that have s as a face, that is,
()
=»
{
}
star
s
t t
Œ
K and
s t
p
.
Definition.
Two simplicial complexes are said to be
combinatorially equivalent
if they
have isomorphic subdivisions.
Definition.
Call a triangulation (K,j) for a topological manifold a
proper triangula-
tion
if the subcomplexes that triangulate the boundary of the stars of vertices are
combinatorially equivalent to the boundary simplicial complex of an n-simplex. A
topological manifold that admits a proper triangulation is called a
combinatorial
manifold
.
The condition for a proper triangulation is stronger than just saying that the stars
are homeomorphic to
D
n
and is basically a generalization of the proper triangulations
defined above for surfaces. In 1952 Moise [Mois52] proved that all topological three-
dimensional manifolds could be triangulated. One of the major achievements of the
1960s was the solution to the triangulation problem by Kirby and Siebenmann
([KirS69]). They proved that in each dimension n, n ≥ 5, there are topological mani-
