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can be interpreted in terms of homology. us, we have the following description of the homology
of a manifold
M
[
90
]:
1
.a;b/
contains only one critical
point
p
of
f
, of index
, and let
'
p
be the attaching map of the
-cell corresponding to
p
. en:
eorem 8.4
Let
a;b
be real numbers such that
a < b
and
f
(a) if
k¤
and
k¤1
then
H
k
.M
b
/H
k
.M
a
/
(b)
H
1
.M
b
/H
1
.M
a
/=ImH
1
.'
p
/
(c)
H
.M
b
/H
.M
a
/KerH
1
.'
p
/
is means that, depending on the attaching map
'
p
, only the homology degrees
1
and
can be affected by the adjunction of a
-cell. In particular, the Betti number
1
can decrease
and
can increase. For example, in the case shown in Figure
8.4
, when we pass through the
critical point
p
according to the increasing value of
f
, a tunnel is created, and
1
increases from
the value 0 to the value 1.
is characterization gives a hint of the ideas underlying Reeb graphs, size theory, topo-
logical persistence and Morse shape descriptors, detailed in chapters
9
,
10
and
11
.
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