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and x and assign a -1 otherwise. If x is a trivial bundle, the Euler number would be
0 because we can move the second copy of M completely off the first so that they do
not intersect at all. In fact, all we need to separate the two copies of M is a nonzero
cross-section, so that we can see that there is a close connection between the Euler
number of a vector bundle and the existence of a nonzero cross-section. In general,
given a cross-section s the zeros of this cross-section will correspond to where s( M )
and s 0 ( M ) will intersect. One can show that the Euler number of x can be determined
from the intersections of s( M ) and s 0 ( M ), no matter which cross-section s one chooses
as long as s( M ) and s 0 ( M ) meet transversally.
8.11.11. Theorem. Let x be an oriented n-plane bundle over a closed compact ori-
ented manifold M n . Then the Euler number of x vanishes if and only if x admits a
nonzero cross-section.
Proof.
See [Hirs76].
Now it is hard to draw pictures that show what is going on because the dimen-
sions get too large; however, we can show something if we drop the hypotheses that
manifolds and bundles are oriented. First, we have to point out that the definitions
above dealing with intersection numbers can be given without any assumptions of
orientability as long as we do not use signed numbers and work modulo 2, that is,
take all values to lie in Z 2 rather than Z . This will give us well-defined mod 2 inter-
section and Euler numbers . (We could also have defined a mod 2 degree of a map
between arbitrary manifolds.) Theorem 8.11.11 would hold for the mod 2 Euler
number without any hypothesis about orientability.
8.11.12. Example. Consider the open Moebius strip line bundle described in
Example 8.9.2 and Figure 8.29. Figure 8.36 shows the total space, which is an open
Moebius strip. A little thought will convince the reader that it is not possible to move
the zero cross-section so that the resulting curve does not intersect the zero cross-
section. The best we can do is reduce the number of intersections to one, as is the
case with the cross-section s 1 in Figure 8.36. Another possible perturbation of the zero
cross-section is cross-section s 2 which has three intersections but also has a mod 2
Euler number of one. This would imply that the bundle does not have a nonzero cross-
section, something we already knew.
An especially interesting case to which Theorem 8.11.11 applies is the tangent
bundle t M of M . Furthermore, because of Theorem 8.10.11 it follows that the Euler
A
C
E
D
B
S 2
B
D
S 1
E
Figure 8.36.
Case where the mod 2 Euler
number equals 1.
C
A
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