Graphics Reference
In-Depth Information
Table 7.2.4.1
Minimal cell decompositions.
X
n 0 ( X )
n 1 ( X )
n 2 ( X )
Orientable surface (genus k)
1
2k
1
Nonorientable surface (genus k)
1
k
1
v 1
v
c 1
c 1
c
1
2
c 2
v
v 2
v 3
c 1
c
3
(a)
(b)
Figure 7.13.
The dunce hat.
This will lead to a cell complex built up from a sequence of cells c 1 ¢, c 2 ¢,..., c k ¢
derived from the c i whose underlying space has the same homotopy type as ΩCΩ. This
approach leads to Table 7.2.4.1 that lists the number of cells in a minimal cell decom-
positions for compact connected surfaces (without boundary).
Will the construction above lead to a minimal cell decomposition? In low dimen-
sions, the answer is yes, but in general, the answer is no. To see why this is so, con-
sider Figure 7.13 which shows part of the steps in the construction of the space called
the dunce hat . Start with the triangle shown in Figure 7.13(a) and identify the three
edges c 1 , c 2 , and c 3 using the orientation of the edges shown by the arrows. Figure
7.13(b) does not yet show the final picture because one still needs to identify the two
edges marked c . The dunce hat is clearly not collapsible because it has no free edges
but one can prove that it is contractible. A consequence is that there are cellular
decompositions so that if one is not careful about choosing the sequence of collapses,
one might not end up with a minimal cell decomposition. This can happen even
in the case of a simple space such as D n . There are cellular decompositions of D n ,
n > 2, with the property that no sequence of collapses will end up with a point. See
[BurM71].
If we look back over the examples we have given of topological spaces since we
started talking about topology in Chapter 5, we can see that, as far as manifolds were
concerned, they have been pretty limited and were restricted to such “standard” spaces
as surfaces, R n , S n , D n , P n , etc. We got the most variety of spaces from surfaces, but
our classification theorems showed that they also fit into simple patterns. Once one
gets above dimension 2, however, things change drastically and a neat classification
is no longer possible. Just so that the reader does not think that there is nothing new
out there and gets bored with all the current examples, we finish this section by defin-
ing a well-known class of three-dimensional manifolds that are quite different from
the ones we have seen up to now. Fortunately, they are relatively easy to describe in
terms of a cell structure.
 
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