11Manifolds and Poincare duality
III Algebraic Topology
11.2 Orientation of manifolds
Similar to the case of vector bundles, we will only work with manifolds with
orientation. The definition of orientation of a manifold is somewhat similar
to the definition of orientation of a vector bundle. Indeed, it is true that an
orientation of a manifold is just an orientation of the tangent bundle, but we
will not go down that route, because the description we use for orientation here
will be useful later on. After defining orientation, we will prove a result similar
to (the first two parts of) the Thom isomorphism theorem.
For a d-manifold M and x ∈ M, we know that
H
i
(M | x; R)
∼
=
(
R i = d
0 i 6= d
.
We can then define a local orientation of a manifold to be a generator of this
group.
Definition
(Local
R
-orientation of manifold)
.
Fr a
d
-manifold
M
, a local
R
-
orientation of M at x is an R-module generator µ
x
= H
d
(M | x; R).
Definition
(
R
-orientation)
.
An
R
-orientation of
M
is a collection
{µ
x
}
x∈M
of
local R-orientations such that if
ϕ : R
d
→ U ⊆ M
is a chart of M, and p, q ∈ R
d
, then the composition of isomorphisms
H
d
(M | ϕ(p)) H
d
(U | ϕ(p)) H
d
(R
d
| p)
H
d
(M | ϕ(q)) H
d
(U | ϕ(q)) H
d
(R
d
| q)
∼
ϕ
∗
∼
∼
∼
ϕ
∗
∼
sends
µ
x
to
µ
y
, where the vertical isomorphism is induced by a translation of
R
d
.
Definition
(Orientation-preserving homeomorphism)
.
For a homomorphism
f
:
U → V
with
U, V ∈ R
d
open, we say
f
is
R
-orientation-preserving if for each
x ∈ U, and y = f(x), the composition
H
d
(R
d
| 0; R) H
d
(R
d
| x; R) H
d
(U | x; R)
H
d
(R
d
| 0; R) H
d
(R
d
| y; R) H
d
(V | y; R)
translation excision
f
∗
translation excision
is the identity H
d
(R
d
| 0; R) → H
d
(R
d
| 0; R).
As before, we have the following lemma:
Lemma.
(i) If R = F
2
, then every manifold is R-orientable.
(ii)
If
{ϕ
α
:
R
d
→ U
α
⊆ M}
is an open cover of
M
by Euclidean space such
that each homeomorphism
R
d
⊇ ϕ
−1
α
(U
α
∩ U
β
) U
α
∩ U
β
ϕ
−1
β
(U
α
∩ U
β
) ⊆ R
d
ϕ
−1
α
ϕ
−1
β
is orientation-preserving, then M is R-orientable.
Proof.
(i) F
2
has a unique F
2
-module generator.
(ii)
For
x ∈ U
α
, we define
µ
x
to be the image of the standard orientation of
R
d
via
H
d
(M | x) H
α
(U
d
| x) H
d
(R
d
| ϕ
−1
α
(x)) R
d
(R
d
| 0)
∼
=
(ϕ
α
)
∗
trans.
If this is well-defined, then it is obvious that this is compatible. However,
we have to check it is well-defined, because to define this, we need to pick
a chart.
If
x ∈ U
β
as well, we need to look at the corresponding
µ
0
x
defined using
U
β
instead. But they have to agree by definition of orientation-preserving.
Finally, we get to the theorem:
Theorem. Let M be an R-oriented manifold and A ⊆ M be compact. Then
(i)
There is a unique class
µ
A
∈ H
d
(
M | A
;
R
) which restricts to
µ
x
∈ H
d
(
M |
x; R) for all x ∈ A.
(ii) H
i
(M | A; R) = 0 for i > d.
Proof.
Call a compact set
A
“good” if it satisfies the conclusion of the theorem.
Claim. We first show that if K, L and K ∩ L is good, then K ∪ L is good.
This is analogous to the proof of the Thom isomorphism theorem, and we
will omit this.
Now our strategy is to prove the following in order:
(i) If A ⊆ R
d
is convex, then A is good.
(ii) If A ⊆ R
d
, then A is good.
(iii) If A ⊆ M, then A is good.
Claim. If A ⊆ R
d
is convex, then A is good.
Let x ∈ A. Then we have an inclusion
R
d
\ A → R
d
\ {x}.
This is in fact a homotopy equivalence by scaling away from x. Thus the map
H
i
(R
d
| A) → H
i
(R
d
| x)
is an isomorphism by the five lemma for all
i
. Then in degree
d
, there is some
µ
A
corresponding to
µ
x
. This
µ
A
is then has the required property by definition
of orientability. The second part of the theorem also follows by what we know
about H
i
(R
d
| x).
Claim. If A ⊆ R
d
, then A is good.
For
A ⊆ R
d
compact, we can find a finite collection of closed balls
B
i
such
that
A ⊆
n
[
i=1
˚
B
i
= B.
Moreover, if
U ⊇ A
for any open
U
, then we can in fact take
B
i
⊆ U
. By
induction on the number of balls
n
, the first claim tells us that any
B
of this
form is good.
We now let
G = {B ⊆ R
d
: A ⊆
˚
B, B compact and good}.
We claim that this is a directed set under inverse inclusion. To see this, for
B, B
0
∈ G
, we need to find a
B
00
∈ G
such that
B
00
⊆ B, B
0
and
B
00
is good and
compact. But the above argument tells us we can find one contained in
˚
B
0
∪
˚
B
00
.
So we are safe.
Now consider the directed system of groups given by
B 7→ H
i
(R
d
| B),
and there is an induced map
lim
−→
B∈G
H
i
(R
d
| B) → H
i
(R
d
| A),
since each
H
i
(
R
d
| B
) maps to
H
i
(
R
d
| A
) by inclusion, and these maps are
compatible. We claim that this is an isomorphism. We first show that this is
surjective. Let [
c
]
∈ H
i
(
R
d
| A
). Then the boundary of
c ∈ C
i
(
R
d
) is a finite sum
of simplices in
R
d
\ A
. So it is a sum of simplices in some compact
C ⊆ R
d
\ A
.
But then
A ⊆ R
d
\ C
, and
R
d
\ C
is an open neighbourhood of
A
. So we can
find a good B such that
A ⊆ B ⊆ R
d
\ C.
Then
c ∈ C
i
(
R
d
| B
) is a cycle. So we know [
c
]
∈ H
i
(
R
d
| B
). So the map is
surjective. Injectivity is obvious.
An immediate consequence of this is that for
i > d
, we have
H
i
(
R
d
| A
) = 0.
Also, if
i
=
d
, we know that
µ
A
is given uniquely by the collection
{µ
B
}
B∈G
(uniqueness follows from injectivity).
Claim. If A ⊆ M, then A is good.
This follows from the fact that any compact
A ⊆ M
can be written as a
finite union of compact
A
α
with
A
α
⊆ U
α
∼
=
R
d
. So
A
α
and their intersections
are good. So done.
Corollary.
If
M
is compact, then we get a unique class [
M
] =
µ
M
∈ H
n
(
M
;
R
)
such that it restricts to µ
x
for each x ∈ M. Moreover, H
i
(M; R) = 0 for i > d.
This is not too surprising, actually. If we have a triangulation of the manifold,
then this [M] is just the sum of all the triangles.
Definition
(Fundamental class)
.
The fundamental class of an
R
-oriented mani-
fold is the unique class [M ] that restricts to µ
x
for each x ∈ M.