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10. MORSE AND MORSE-SMALE COMPLEXES
function
f
over
M
(see chapter
8
, definition
8.1
). Indeed, integral lines are pair-wise disjoint,
that is, if their images share a point, then they are the same line. e images of integral lines cover
the whole
M
, but if we consider the integral lines associated with the critical points of
f
, their
images define a partition of
M
.
is partition is used to decompose
M
into regions of uniform flow, thus capturing the
characteristics of the gradient field. More precisely, the
descending manifold
of a critical point
p
is the set
D.p/
of points that flow towards
p
, and the
ascending manifold
of
p
is the set
A.p/
of
points that originate from
p
¹
. In formulae:
A.p/Dfq2MW
lim
t!C1
q
.t/Dpg
t!1
q
.t/Dpg;
where
q
is the integral line at the point
q
. Note that the descending manifold of
f
is the ascend-
ing manifold of
f
.
e descending manifold of a critical point
p
of index
i
is an open
i
-cell. Similarly, the
ascending manifold of a critical point of index
i
is an
ni
open cell. For example, if
M
is a
2
-manifold the descending manifold of a maximum is an open disk, that of a saddle is an open
interval, and that of a minimum is the minimum itself.
e collection of all descending manifolds forms a complex, called the
descending Morse
complex
, and the collection of all ascending manifolds also form a complex, called the
ascending
Morse complex
, which is dual with respect to the descending complex (see Figure
10.1
left and
middle). For instance, when
M
is a
2
-manifold, the
2
-cells of the descending Morse 2-complex
correspond to the maxima of
f
, the
1
-cells to the saddle points, and the 0-cells to the minima.
Symmetrically, the
2
-cells of the ascending Morse 2-complex correspond to the minima of
f
,
the
1
-cells again to the saddle points, and the 0-cells to the maxima. When
M
is a
3
-manifold,
the
3
-cells of a descending Morse
3
-complex correspond to the maxima, the
2
-cells to the
2
-
saddles the
1
-cells to the
1
-saddles, and the
0
-cells to the minima. Symmetrically, the
3
-cells
of the ascending Morse
3
-complex correspond to the minima, the
2
-cells to the
1
-saddles, the
1
-cells to the
2
-saddles, and the
0
-cells to the maxima.
e function
f
is a
Morse-Smale
function if the descending and ascending Morse com-
plexes intersect only transversally.
²
In 2D, this means that, if an ascending 1-manifold intersects
a descending 1-manifold transversally, they cross at exactly one point. is condition implies that
the topological behavior of the images of the integral lines does not change under small pertur-
bations of the vector field [
156
].
In the case of Morse-Smale functions, the
Morse-Smale complex
is defined as the intersec-
tion of the ascending and descending manifolds. e cells of the
Morse-Smale complex
are the
D.p/Dfq2MW
lim
¹In the mathematical literature, the term
unstable
is used instead of ascending, and the term
stable
is used instead of descending
[
156
].
²By definition, two submanifoldsAandBof a manifoldMintersect transversally inpifT
p
ACT
p
BDT
p
MwhereT
p
is
the tangent space atp.
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