Graphics Reference
In-Depth Information
Section 7.2.4
7.2.4.1.
Describe a minimal cell decomposition for the Klein bottle.
7.2.4.2.
Triangulate the dunce hat and compute its homology groups.
7.2.4.3.
(a)
Prove that the homology groups of the lens spaces are what they were stated to
be.
(b)
Prove that the Euler characteristic of a lens space is 0.
Section 7.2.5
7.2.5.1.
Compute the incidence matrices for the simplicial complex K =∂<
v
0
v
1
v
2
v
3
>. Work
through the proof of Theorem 7.2.5.3 and determine the normalized form of the inci-
dence matrices and the bases of the chain groups that define them. Show how these
matrices determine the known homology groups.
Section 7.2.6
7.2.6.1.
(a)
If
X
is a point, then prove that
(
)
ª
HG
H
X
X
;
;
G, and
0
0
(
G
)
=
,
for q
π
0
.
q
(b)
Let
X
be a polyhedron. Prove that H
0
(
X
;G) is isomorphic to a direct sum of as
many copies of G as there are components of
X
. In particular, the 0th connec-
tivity number of
X
, k
0
(
X
), is nothing but the number of components of
X
.
7.2.6.2.
Compute H
q
(
X
;
Z
2
), for all q, where
X
is
S
n
(a)
S
1
¥
S
1
(b)
P
2
(c)
(d)
the Klein bottle
Section 7.3
Prove that the space
X
=
S
2
⁄
S
1
⁄
S
1
in Figure 7.15 has the same homology groups as
7.3.1.
the torus.
Section 7.4.1
7.4.1.1.
Complete the proof of Theorem 7.4.1.6 by filling in all missing details.
7.4.1.2.
Prove Theorem 7.4.1.7.
7.4.1.3.
Prove Lemma 7.4.1.11.
7.4.1.4.
Prove Theorem 7.4.1.12.
