Graphics Reference
In-Depth Information
A decomposition of
S
2
for which Euler's
theorem fails.
Figure 6.28.
v
3
v
4
v
1
v
2
v
7
v
6
v
8
v
5
6.7
E
XERCISES
Section 6.1
6.1.1.
The proof of Euler's formula (Theorem 6.1.1) used the fact that the triangular faces in
the decomposition of a disk
D
2
can be listed in a sequence
T
1
,
T
2
,...,
T
k
, such that
T
i
meets
X
=
U
T
i
-
1
j
1
£<
ji
in either one or two edges. Sketch a proof and discuss potential problems. (In [BurM71]
it is proved that the analogous fact for cell decompositions of higher-dimensional disks
does not hold.)
6.1.2.
A common way to express Euler's theorem is to say that no matter how a sphere is
divided into n
f
regions with n
e
edges and n
v
vertices, the sum n
v
- n
e
+ n
f
will always
equal 2. Compute n
v
- n
e
+ n
f
for the decomposition of
S
2
shown in Figure 6.28. Where
does the proof of Euler's theorem fail in that example? If we want to preserve the valid-
ity of the theorem, then what conditions must a “permissible region” satisfy so that
Euler's theorem will hold for all decompositions of
S
2
into permissible regions?
Section 6.3
6.3.1.
Let K be a simplicial complex and let
x
ŒÔKÔ. Prove that there is a unique simplex
s ΠK such that
x
Πint s.
6.3.2.
Show that if L and M are subcomplexes of a simplicial complex K, then L « M is a sub-
complex of K.
6.3.3.
Prove that if K is a simplicial complex, then so is ∂K.
6.3.4.
Let K be a simplicial complex.
(a)
Prove that K is connected if and only if ÔKÔ is connected.
(b)
Define a
component
of K to be a maximal connected subcomplex L of K. Show that
KL L
=»»»
K
L
n
,
1
2
