Graphics Reference
In-Depth Information
It is worthwhile pointing out the following properties of this CW complex repre-
sentation of
P
n
explicitly:
(1) There is one i-dimensional cell for each dimension 0 £ i £ n.
(2) The i-skeleton of
P
n
is just
P
i
and we have a filtration
=Ã
(
)
ÃÃ
0
0
1
1
n
-
1
n
PcPS
...
P
Ã
P
,
where
P
i
is obtained from
P
i-1
by attaching an i-cell.
This cell structure of
P
n
allows us to compute the homology and cohomology of the
space fairly easily if we use the approach based on oriented cells described in Section
7.2.4. The fact is that each i-cell
c
1
in
S
i
has a natural orientation obtained by pro-
jecting the standard orientation of
D
i
upward. The projection p projects this orienta-
tion to an orientation of the cell
c
i
in
P
n
. If we denote our CW complex for
P
n
by C,
then since there is only one cell in each dimension i, 0 £ i £ n,
i
()
ª
Z
CC
and to compute the homology groups we simply have to analyze the boundary maps
on the cells
c
i
.
The homology groups of
P
n
are given by
7.7.1. Theorem.
(1) H
0
(
P
n
) ª
Z
0
for
for
0 < i < n and i even
0 < i < n and i o
()
ª
Ó
˛
n
H
P
i
Z
dd
2
0
n
ndd
even
o
()
ª
Ó
˛
n
H
P
n
Z
(2) H
i
(
P
n
,
Z
2
) ª
Z
2
, 0 £ i £ n,
Proof.
Here is a sketch of the argument that proves (1). Let us orient the cells
c
i+1
based on the orientation of
D
i+1
induced from the standard orientation [
e
1
,
e
2
,...,
e
i+1
]
of
R
i+1
. Consider the cell
c
i+1
in
S
n
with i ≥ 2. The boundary of that cell consists of
the two cells
c
i
1
and
c
i
2
. What orientation does the boundary map on
c
i+1
induce on
these two cells? Well, the orientation induced on
c
i
1
and
c
i
2
have to be [
e
1
,
e
2
,...,
e
i
]
and [-
e
1
,
e
2
,...,
e
i
], respectively. The reason is that adding
e
i+1
at the end of the first
basis and -
e
i+1
at the end of the second must lead to the standard orientation of
D
i+1
and it is easy to check in the second case that
[
]
=
[
]
-
ee
,
,...,
e e
,
-
ee
,
,...,
e
.
12
i
i
+
1
12
i
+
1
But under the antipodal identification map the cell
c
i
2
and its orientation [-
e
1
,
e
2
,
...,
e
i
] gets mapped to the cell
c
i
1
with orientation [
e
1
,-
e
2
,..., -
e
i
]. The latter agrees
