Graphics Reference
In-Depth Information
Definition.
The
index of f at
p
with respect to
N
, I
p
(f,
N
), is defined by
I
p
= (f,
N
) =+1,
if the composite
()
æÆ
()
=≈
(
)
()
T
M
ææ
T
W
t
n
æÆ
ææææ
n
(
)
p
f
p
N
N
(
)
N
(
)
f
p
f
p
Df
(
p
)
projection
is orientation preserving (the composite is an isomorphism
because of our transversality hypothesis)
= -1,
otherwise.
The
intersection number of f with respect to
N
, I(f,
N
), is defined by
Â
-
1
(
)
=
(
)
(
)
π
IN
f
,
I
f
,
N
,
if f
N
f
.
p
pN
-
1
()
Œ
f
=
0
,
otherwise
.
8.11.10. Theorem.
If f, g :
M
Æ
W
are homotopic differentiable maps that are both
transverse to
N
, then I(f,
N
) = I(g,
N
).
Proof.
See [Hirs76].
By Theorem 8.11.10, the next definition is well defined.
Definition.
Given an arbitrary continuous map f :
M
Æ
W
, let g :
M
Æ
W
be any dif-
ferentiable map that is homotopic to f and transverse to
N
. Define the
intersection
number of f with respect to
N
, I(f,
N
), to be I(g,
N
).
Definition.
If
M
n
is actually a submanifold of
W
n+k
and if i :
M
Õ
W
is the inclusion
map, then define the
intersection number of
M
n
and
N
k
in
W
n
+
k
, I(
M
,
N
), to be I(i,
N
).
If
M
is transverse to
N
and if
p
Œ
M
«
N
, then I
p
(i,
N
) is called the
index of the inter-
section
of
M
and
N
at
p
.
The intersection number I(
M
,
N
) of the two submanifolds of “dual” dimensions n
and k in the (n + k)-dimensional manifold
W
is what is most interesting to us here.
Compare this with the intersection numbers in Section 7.5.2. The two concepts are
closely related. We can carry things a step further to get an intersection number of a
single manifold with “itself.”
Definition.
If x is an oriented n-plane bundle over a closed compact oriented man-
ifold
M
n
, then define the
Euler number
of x, I(x), to be I(s
0
(
M
),s
0
(
M
)), where s
0
is the
zero cross-section of x.
Here is a way to visualize the Euler number of the oriented vector bundle x. Think
of having two copies of
M
sitting at the zero cross-section in the 2n-dimensional total
space of x. The transversality theorem implies that we can move the second copy of
M
slightly so that it meets the first copy transversally. The Euler number is then gotten
by assigning a +1 or -1 to each intersection and adding these together. We assign
a +1 at an intersection if the orientation of the two copies of
M
induce the same
orientation as the orientation induced on the total space of x by the orientation of
M
