Graphics Reference
In-Depth Information
Actually, a stronger result holds in this case stating that
M
a
and
M
b
are diffeomorphic,
namely, there is a differentiable and invertible function between
M
a
and
M
b
, whose inverse is
also differentiable (see Figure
8.4
(b)).
Let
p
be a critical point of
f
with index
and let
f.p/Dc
. en, for each
"
such
eorem 8.3
1
.c";cC"/
contains no critical points other than
p
, the set
M
cC"
has the same homotopy
type of the set
M
c"
with a
-cell attached:
that
f
M
cC"
M
c"
[
'
p
e
:
According to the definitions in section
7.1
, the
attaching map
'
p
identifies each point
x2
S
1
with the point
'
p
.x/2M
c"
(see Figure
8.4
(c)).
In order to study the changes in the level sets
V
t
Df
1
.t/
, an approach to Morse theory
based on the attaching of handles [
142
], rather than cells, can be used, as in [
89
] for the case of
surfaces. When
f
is defined on a surface, if
t
is a regular value for
f
then
V
t
, if not empty, is the
union of finitely many smooth circles. Moreover, if
a;b
are real numbers such that
a < b
, then
1.
if the set
f
1
.a;b/
contains no critical points for
f
, then
V
a
and
V
b
are diffeomorphic;
1
.a;b/
contains only one critical point of index
0
for
f
, then
V
b
is the union
of
V
a
with a circle;
2.
if the set
f
3.
if the set
f
1
.a;b/
contains only one critical point of index
2
for
f
, then
V
b
is diffeomor-
phic to
V
a
without one of its circles;
4.
if the set
f
1
.a;b/
contains only one critical point of index
1
for
f
, then the number
of connected components of
V
b
differs from that of
V
a
by
1
,
0
or
1
depending on the
attaching map.
In case (4), the difference in the number of connected components is non-zero if the handle (in
this case, the strip
0;10;1
) is attached without twists (or with an even number of twists),
while it is
0
if there is an odd number of twists. e presence of an odd number of twists implies
that the surface is non-orientable. erefore, when the surface is embedded in
R
3
,
V
a
and
V
b
necessarily have a different number of connected components.
8.3 HOMOLOGY OF MANIFOLDS
Morse theory asserts that changes in the topology of a manifold endowed with a Morse function
occur in the presence of critical points; since most manifolds can be triangulated as simplicial
complexes and a Morse function can be discretized on simplices, those changes in the topology
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