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not want to define continuously varying families of orientations of tangent spaces or
normal vector fields. To prove the compatibility of our two definitions of orientabil-
ity one might be tempted to take a direct approach and show, for example, that
orientations of tangent spaces induce coherent orientations of the simplices in a
triangulation. Unfortunately, that would be technically complicated because homeo-
morphisms do not preserve any vector space structure. The better approach is to prove
the next theorem from the definitions without reference to pseudomanifolds, using
only properties of homology groups.
8.5.3. Theorem.
A closed compact connected n-dimensional differentiable mani-
fold
M
n
is orientable (according to the definitions in this section) if and only if
H
n
(
M
n
) ª
Z
.
Proof.
Conceptually the proof is not hard and consists of two steps:
Step 1:
Relate an orientation of a tangent plane at a point to an orientation of
a neighborhood of the point in
M
.
Step 2:
Relate a “continuously varying” collection of local orientations to a
homology class.
Although we have not defined what is meant by the orientation of a neighborhood
of a point it should be at least intuitively obvious. Think back to our discussion of the
orientation of a surface. It is the old story. We have a concept in the linear setting of
vector spaces and we extend it to curved spaces via a linearization process, that is, we
use tangent planes to approximate the space locally. Filling in the details for Step 1
would be very messy given our current approach to manifolds. To do things more ele-
gantly involves a more abstract approach that we cannot go into here. One would need
to know about vector bundles (defined in Section 8.10) and more. More details for
Step 1 can be found in [MilS74] or [Span66].
Step 2 is the more straightforward part and not that hard, but we are not able to
present it here because it uses properties of homology groups we did not state or prove
in Chapter 7. Roughly speaking, one shows that an orientation of
M
is equivalent to
a choice of generator
(
)
ª
m
p
Œ
H
n
MM p
,
-
Z
for each
p
Œ
M
that “varies continuously” in the sense that for some compact neigh-
borhood
N
of
p
there is a class
(
)
m
N
Œ
H
n
MM N
,
-
that “restricts” to m
p
. The existence of such generators leads to a unique nonzero
element
(
)
n
m
M
Œ
H
n
M
called the
fundamental homology class
of the oriented manifold that is a generator for
H
n
(
M
n
). Again see [MilS74] or [Span66].
