Graphics Reference
In-Depth Information
Proof.
See [Stee51] or [Span66]. The isomorphisms in (3) are induced by the
projection map in (7.17).
7.8
E
XERCISES
Section 7.2.1
7.2.1.1.
Prove Lemma 7.2.1.2.
7.2.1.2.
Prove that if K is a simplicial complex, then
rank (H
0
(K)) = number of connected components of ΩKΩ.
7.2.1.3.
Prove that the homology groups of the Klein bottle are as indicated in Table 7.2.1.1.
You can use a triangulation similar to that of the torus shown in Figure 7.5.
7.2.1.4.
Prove the results indicated in Table 7.2.1.1 for
(a)
orientable surfaces of genus k
(b)
nonorientable surfaces of genus k
7.2.1.5.
If
X
and
Y
are polyhedra, prove that
(
)
ª
()
≈
()
H
XY
⁄
H
X
H
Y
,
q
π
0
q
q
q
Section 7.2.2
7.2.2.1.
Prove Lemma 7.2.2.6.
7.2.2.2.
Let K =∂<
v
0
v
1
v
2
v
3
> be the simplicial complex in Example 7.2.1.6. Compute the maps
f
*q
for the simplicial map f : K Æ K defined by f(
v
0
) =
v
0
, f(
v
1
) =
v
2
, f(
v
2
) =
v
1
, and
f(
v
3
) =
v
3
.
7.2.2.3.
Let K be a simplicial complex. Show that sd(K) is also a simplicial complex.
Section 7.2.3
7.2.3.1.
Let K be a simplicial complex. Show that the cone on ΩKΩ is a polyhedron and
determine its homology groups.
7.2.3.2.
If K is a nonempty simplicial complex and if ΩKΩ has k components, prove that the
suspension of ΩKΩ has the following homology groups:
(
)
=
HSK
HSK k
HSKHKq
Z
0
(
)
ª-
(
)
1
Z
1
(
)
ª
()
,
>
0
.
q
+
1
q
