Graphics Reference
In-Depth Information
M
s
-1
(
[
]
)
=
f
s
,
•
.
If we define
gnf
=-
:
MR
Æ
,
then note that
M
s
, as defined for f, is the same as
M
n-s
, as defined for g.
Now let
p
be a critical point of index k for f and assume that it is the only criti-
cal point with critical value c = f(
p
). Assume that f has no critical values in [c -e,c +
e] other than c. Let
-
1
(
[
]
)
A
=
f
c c
-
e
,
+
e
.
This set is diffeomorphic to
D
k
¥
D
n-k
and is the set that
M
c-e
and
M
c+e
have in
common. It follows from what we did above that
c
-
e
c
+
e
k
n
-
k
MM
=
»
DD
¥
knk
--
1
D
¥
S
and
k
n
-
k
MM
=
»
DD
SD
¥
.
c
+
e
c
-
e
k
-
1
n
-
k
¥
See Figure 8.24 and compare this to Figure 8.21. To put it another way,
M
c-e
has
the same homotopy type as
M
c+e
with an n - k cell attached and
M
c+e
has the
homotopy type of
M
c-e
with a k cell attached. Since we get the same space
whether we start building it from the top or the bottom, what this shows is a
fundamental duality between k and n - k cells. This is the Poincaré duality in Section
7.5.2.
M
c-Œ
M
c-Œ
M
c+Œ
P
p
D
k
¥ S
n-k-1
D
k
¥ D
n-k
Figure 8.24.
The dual handle decomposition.
