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In-Depth Information
Figure 8.35.
Making sense of
deg(f,
q
).
f(A
3
)
A
2
A
1
p
2
p
1
f(A
1
)
q
A
3
p
3
f(A
2
)
f
S
1
S
1
f(S
1
)
8.11.8. Theorem.
(1) The value deg(f,
q
) does not depend on the choice of regular value
q
.
(2) If f, g :
M
Æ
N
are homotopic differentiable maps and if
q
and
q
¢ are regular
values for f and g, respectively, then deg(f,
q
) = deg(g,
q
¢).
Proof.
See [Miln65b] or [Hirs76].
Theorem 8.11.8 means that the next two definitions are well defined.
Definition.
The
degree of f
, deg f, is defined to be deg(f,
q
) for any regular value
q
for f.
Definition.
If g :
M
Æ
N
is any continuous map, then define the
degree of g
, deg g, to
be the degree of any differentiable map f :
M
Æ
N
that is homotopic to g.
8.11.9. Theorem.
The definition of the degree of a map above agrees with the def-
inition of the degree of a map given in Section 7.5.1.
Proof.
The proof is not hard, but the best way to prove this theorem is to use the
tangent bundle of a manifold, its relationship to orientability, and the connection
between that and the top-dimensional homology group of the manifold that we dis-
cussed in earlier sections.
Next, we use transversality to help shed more light on the duality in manifolds
that we discussed in Section 7.5.2. Let
N
k
be a closed submanifold of a manifold
W
n+k
.
We shall identify t
N
≈n
N
with t
W
|
N
and assume that the normal n-plane bundle n
N
is
oriented with orientation m. Let (
M
n
,s) be a closed compact oriented manifold. Con-
sider a differentiable map
n
f
:
MW
Æ
that is transverse to
N
. We know that f
-1
(
N
) consists of a finite number of isolated
points. Let
p
Πf
-1
(
N
).
