Orientation of manifolds
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1 Zero dimensional manifolds
For zero dimensional manifolds an orientation is a map from the manifold to , i.e. an orientation is a map
. From now on we assume that all manifolds have positive dimension. Unless otherwise stated the manifolds have empty boundary.
2 Orientation of topological manifolds
An orientation of a topological manifold is a choice of a maximal atlas, such that the coordinate changes are orientation preserving. To make this precise we have to define when a homeomorphism from an open subset of
to another open subset
is orientation preserving. We do this in terms of singular homology groups.
Definition 2.1. A homeomorphism from an open subset
of
to another open subset
is orientation preserving, if for each
the map
is the identity map. Here the isomorphisms
is the following: We first take the map to
(or to
) induced by the translation mapping
to
resp.
and then the inverse of the excision isomorphism. The isomorphism
is the inverse of the corresponding map.
Definition 2.2. An orientation of an -dimensional topological manifold
is the choice of a maximal oriented atlas. Here an atlas
is called oriented if all coordinate changes
are orientation preserving.
Note that any oriented atlas defines a maximal oriented atlas by adding all charts such that the atlas is still oriented. This is normally the way an oriented atlas is given.
A topological manifold is called orientable if it has a topological orientation, otherwise it is called non-orientable.
A topological manifold together with a topological orientation is called an oriented topological manifold.
An open subset of an oriented topological manifold is oriented by restring the charts in the maximal oriented atlas to the intersection with the open subset. A homeomorphism between oriented topological manifolds is orientation preserving if for each chart
in the oriented atlas of
the chart
is in the oriented atlas of
.
There are several equivalent formulations of orientations both for topological manifolds and for smooth manifolds which we will explain in the following sections.
3 Reformulation in terms of local homological orientations
An orientation of an -dimensional topological manifold
can also be defined in terms of the local homology groups
for each
in
.
Definition 3.1. A local homological orientation of an -dimensional topological manifold
is the choice of a generator
of the local homology group
for each
. Such a choice is called continuous, if for each
there is an open neighborhood
and a class
such that the map induced by the inclusion
maps
to
for each
. A homological orientation of
is a continuous choice of local homological orientations.
As above an open subset of
has an induced homological orientation which is given by the image under the inverse of the isomorphism induced by the inclusion
.
To get an example consider a finite dimensional oriented real vector space , i.e.
is equipped with an equivalence class of bases
, where two bases are called equivalent, if and only if the matrix of the base change matrix has positive determinant. The orientation of
as a vector space gives a homological orientation of
as a topological space as follows. We first orient at
in
by considering the simplex spanned by
. This contains
in its interior and is a generator of
. By translations we define local orientations at arbitrary points of
mapping the local orientation at
to the local orientation at
by the map induced by the translation mapping
to
. By construction this is a continuous family of local homological orientations and so gives a homological orientation of
. From this we obtain homological orientations of all open subsets of
.
The equivalence of these two concepts of an orientation of a topological manifold is shown as follows. A homeomorphism between manifolds equipped with a continuous family of local orientations is called orientation preserving if the induced map maps the corresponding local orientations to each other. We note that if both manifolds are open subsets of , this definition of orientation preserving homeomorphisms agrees with the one defined above. With this one defines for a topological manifold with a continuous family of local orientations a maximal oriented atlas by all charts which are orientation preserving, where we orient
as above. In turn if one has a maximal oriented atlas one uses it to transport the local orientations of open subsets of
to local homological orientations of
, which are a continuous family, since the atlas is oriented.
4 Orientation of smooth manifolds
The definition of an orientation for a topological manifold needs homology groups. For smooth manifolds the definition can be simplified. To distinguish the very similar definition we call it a smooth orientation.
Definition 4.1. A smooth orientation of an -dimensional smooth manifold
is the choice of a maximal oriented atlas. Here a smooth atlas
is called oriented if the determinant of the derivatives of all coordinate changes
is positive. The oriented atlas is called maximal if it cannot be enlarged to an oriented atlas by adding another chart. Note that any oriented atlas defines a maximal oriented atlas by adding all charts such that the atlas is still oriented. This is normally the way an oriented atlas is given.
A smooth manifold is called orientable if it has a smooth orientation, otherwise it is called non-orientable.
A smooth manifold together with a smooth orientation is called an oriented smooth manifold.
5 From tangential orientations to homological orientations
For smooth manifolds we have now two definitions of an orientation, the smooth orientation and the orientation as a topological manifold. Here we explain why they are again equivalent concepts. The key observation is the following. If we have an orientation of the vector space we have defined corresponding local orientations. If we change the orientation of the vector space
, the local homological orientation changes its sign. Since there are two orientations of
as a vector space and two generators of
this correspondence is a bijection.
The next observation is that if we have a diffeomorphism from an open subset in
to another open subset
in
, then its differential preserves the standard orientation of
if and only if it preserves the corresponding local homological orientations and so the underlying homeomorphism is orientation preserving in the sense defined in the beginning of the last section.
Thus, if is smoothly oriented, i.e. is equipped with a maximal oriented smooth atlas, then - forgetting the smooth structure - we obtain an oriented topological atlas and we define the corresponding topological orientation by passing to the maximal oriented topological atlas containing these charts. In turn, if one has a maximal oriented topological atlas the subset of smooth charts in it defines a smooth orientation.
6 Reformulations of orientation for smooth manifolds
There are several equivalent formulations for orientations of smooth manifolds.
Definition 6.1. A tangential orientation of is a continuous choice of an orientation of the tangent space
in the sense of orientations of vector spaces for every point
. Here continuous means that for every
M there is a chart
around
, such that the differential of
maps for all
the orientation at
to the same orientation of
.
The relation between these two definitions is the following. If is an oriented atlas, we define an orientation of
by choosing an oriented chart
around
and define the tangential orientation as the image of the orientation of
under the differential of
. In turn, if a continuous orientation of
for all
is given, one defines a maximal oriented atlas as the atlas consisting of all charts
such that all differentials are orientation preserving, where we equip
with the induced orientation from
equipped with the standard orientation given by the canonical basis. It is easy to check that these constructions are well defined and give equivalent formulations.
Further equivalent formulations, which need a bit more knowledge of vector bundles are:
- An orientation of a smooth n-dimensional manifold is given by the reduction of the structure group
of the tangent bundle
to
, the subgroup of matrices with determinant
. That this is an equivalence is an easy exercise.
- An orientation of a smooth manifold is given by a trivialization (an isomorphism to the trivial bundle) of the exterior bundle . That this is an equivalence is an easy exercise.
Remark 6.2. Since the different concepts of orientations are all equivalent, one normally speaks of an oriented manifold in all cases. We only used the adjectives to make clear that a priory the definitions are different.
7 Criteria for orientability
There are various criteria for orientability:
Theorem 7.1. A smooth -dimensional manifold is orientable if and only if the tangent bundle (or the normal bundle of an embedding into
) has a Thom class, i.e. a class
, whose restriction to each fibre
is a generator of
. Moreover the choice of a Thom class determines an orientation and vice versa.
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![TM_x](/images/math/b/8/b/b8b036f2add61069375cfe6ac8f65851.png)
![TM](/images/math/8/e/4/8e4f2fdbb13a819d0f3d6796804f9ef4.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 7.2. A smooth manifold is orientable if and only if the first Stiefel Whitney class of its tangent bundle vanishes.
See [Milnor&Stasheff1974, Lemma 11.6 and Problem 12-A] and [Bredon1993, Proposition 17.2].
Theorem 7.3 [Dold1995, VIII Corollary 3.4]. A connected closed -dimensional manifolds
is
orientable if and only if
is non-zero, in which case it
is isomorphic to
. The choice of a generator is called a fundamental class
. The choice of
a generator corresponds to the choice of an orientation [Dold1995, VIII Definition 4.1]. For
a not necessarily connected compact oriented manifold
the components
are oriented and the sum of the fundamental classes of the components
define the fundamental class of
.
There is a generalization of Theorem 7.3 to non-compact manifolds.
Theorem 7.4 [Greenberg&Harper1981, Corollary 22.26]. If is arbitrary, then
is orientable if and only if for each
compact connected subset
there is a class
, such that for each
the map induced by the inclusion
maps
to a generator of
and the classes
mapped to each other under the maps induced by the inclusion
for all
.
The images of the classes
in
define a
homological orientation of
and in turn a homological orientation
determines the classes
.
8 Manifolds with boundary
For manifolds with boundary an orientation is defined as an orientation of its interior. An orientation of
induces an orientation on the boundary
. If
is
-dimensional, we orient the boundary, which is
-dimensional by attaching to
the local orientation
, if the restriction of a chart around
from
to
to the interior is in the oriented atlas of the interior of
. Otherwise we define
. For example it we orient the interval
by the atlas of the interior given by the identity map, then
and and
.
If the dimension of is positive, we define the induced orientation both for smooth or topological manifolds in terms of an induced maximal oriented atlas of the boundary. If
is a (smooth) chart around a boundary point
, such that its restriction to the inner is in the oriented atlas of the inner of
, then the restriction of this chart to
is a chart of
and these charts form a maximal oriented (smooth) atlas of
. The orientation given by this atlas is called the induced orientation on
.
The convention, that we consider the negative orientation on the boundary is for smooth manifolds equivalent to choosing an identification of the restriction of the tangent bundle of to
with
, where we identify
with a subbundle by selecting the outward normal vector field. With other words for smooth manifolds the induced orientation is characterized as the orientation of
, such that any outward pointing normal vector plus this orientation is the given orientation of
.
As for compact manifolds without boundary one can see that a compact connected manifold with boundary is orientable if and only if
is non-zero, in which case it is again isomorphic to
, [Dold1995]. The choice of a generator is called a relative fundamental class and again this fixes an orientation of
.
Our at the first glance slightly ad libitum looking convention is made in such a way that the following holds:
Theorem 8.1. Let be a compact oriented
-dimensional manifold with boundary. If
is the fundamental class compatible with the orientation, then
is the the fundamental class compatible with the induced orientation of the boundary as defined above.
Since the proof of this result is not in standard text books (to my knowledge), we give it here.
Proof. Since the orientation is given locally (we use the homological formulation) it is enough to show that if we consider the local orientation of in a chart near the boundary, the boundary operator maps it to the local orientation of
in the restriction of this chart to the boundary. Here we choose the chart in such a way, that the orientation of
corresponds to the standard orientation of
(if not change your atlas by a reflection in
).
Thus we consider and the local orientation given by the standard basis of
. Since we work with the half space we map the simplex constructed by the standard basis with edges
, so that it is spanned instead by
. We denote this simplex by
. The class represented by this simplex in
for some
in the inner of the simplex is the same as that of
given by the standard orientation of
. If we begin with the fundamental class
, consider its image under the boundary operator in
and pass to the local orientation at
, then it is represented by the restriction of
to the boundary of
. More precisely
, where
corresponds to the simplex spanned by the corresponding vectors, and the local orientation in
at
corresponding to the image of the fundamental class in
is given by
. But this is the negative of the local orientation of
given by the standard basis. This finishes the proof and explains why we took the negative orientation in our construction of the induced orientation in terms of an atlas.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
9 Orientation of products
Given two oriented manifolds there is an obvious way to orient their product by choosing the product atlas. If is smooth and we have given orientations as tangential orientations, we note that
is isomorphic to
and the isomorphism is induced by the differential of the projections and then the product orientation is given by the juxtaposition of the orientations of
and
.
Similarly if and
are oriented by a continuos family of local homological orientations, we note that
is isomorphic to
, this isomorphism from the latter to the first is given by the cross product
. By definition of the cross product of the local homological orientation given by the standard basis of
with the local homological orientation given by the standard basis of
is the local homological orientation given by the standard basis of
. Thus the different concepts of product orientations given by the product of an atlas and by the product of local homological orientations agree also.
As a consequence for compact oriented manifolds equipped with fundamental classes the cross product of the fundamental classes corresponds to the product of the orientations induced by the fundamental classes.
10 Orientation of complex manifold
An -dimensional complex manifold is a topological manifold together with an atlas
such that the coordinate changes are holomorphic maps. Given such an atlas the charts considered as maps to
have orientation preserving coordinate changes, since a complex matrix considered as a real matrix has determinant
, the square of the norm of the complex determinant. Thus a complex manifold considered as a real manifold has this way a canonical orientation.
11 References
- [Bredon1993] G. E. Bredon, Topology and geometry, Graduate Texts in Mathematics 139, Springer-Verlag, 1993. MR1224675 (94d:55001) Zbl 0934.55001
- [Dold1995] A. Dold, Lectures on algebraic topology, Springer-Verlag, 1995. MR1335915 (96c:55001) Zbl 0872.55001
- [Greenberg&Harper1981] M. J. Greenberg and J. R. Harper, Algebraic topology, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, 1981. MR643101 (83b:55001) Zbl 0498.55001
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
12 External links
- The Encylopedia of Mathematics article on orientation.
- The Wikipedia page on orientability.
- The Wikipedia page on orientations of vector spaces.
![\epsilon : M \to \{ \pm 1 \}](/images/math/4/6/c/46c2b13b508393e2ed97a260b9023207.png)
2 Orientation of topological manifolds
An orientation of a topological manifold is a choice of a maximal atlas, such that the coordinate changes are orientation preserving. To make this precise we have to define when a homeomorphism from an open subset of
to another open subset
is orientation preserving. We do this in terms of singular homology groups.
Definition 2.1. A homeomorphism from an open subset
of
to another open subset
is orientation preserving, if for each
the map
is the identity map. Here the isomorphisms
is the following: We first take the map to
(or to
) induced by the translation mapping
to
resp.
and then the inverse of the excision isomorphism. The isomorphism
is the inverse of the corresponding map.
Definition 2.2. An orientation of an -dimensional topological manifold
is the choice of a maximal oriented atlas. Here an atlas
is called oriented if all coordinate changes
are orientation preserving.
Note that any oriented atlas defines a maximal oriented atlas by adding all charts such that the atlas is still oriented. This is normally the way an oriented atlas is given.
A topological manifold is called orientable if it has a topological orientation, otherwise it is called non-orientable.
A topological manifold together with a topological orientation is called an oriented topological manifold.
An open subset of an oriented topological manifold is oriented by restring the charts in the maximal oriented atlas to the intersection with the open subset. A homeomorphism between oriented topological manifolds is orientation preserving if for each chart
in the oriented atlas of
the chart
is in the oriented atlas of
.
There are several equivalent formulations of orientations both for topological manifolds and for smooth manifolds which we will explain in the following sections.
3 Reformulation in terms of local homological orientations
An orientation of an -dimensional topological manifold
can also be defined in terms of the local homology groups
for each
in
.
Definition 3.1. A local homological orientation of an -dimensional topological manifold
is the choice of a generator
of the local homology group
for each
. Such a choice is called continuous, if for each
there is an open neighborhood
and a class
such that the map induced by the inclusion
maps
to
for each
. A homological orientation of
is a continuous choice of local homological orientations.
As above an open subset of
has an induced homological orientation which is given by the image under the inverse of the isomorphism induced by the inclusion
.
To get an example consider a finite dimensional oriented real vector space , i.e.
is equipped with an equivalence class of bases
, where two bases are called equivalent, if and only if the matrix of the base change matrix has positive determinant. The orientation of
as a vector space gives a homological orientation of
as a topological space as follows. We first orient at
in
by considering the simplex spanned by
. This contains
in its interior and is a generator of
. By translations we define local orientations at arbitrary points of
mapping the local orientation at
to the local orientation at
by the map induced by the translation mapping
to
. By construction this is a continuous family of local homological orientations and so gives a homological orientation of
. From this we obtain homological orientations of all open subsets of
.
The equivalence of these two concepts of an orientation of a topological manifold is shown as follows. A homeomorphism between manifolds equipped with a continuous family of local orientations is called orientation preserving if the induced map maps the corresponding local orientations to each other. We note that if both manifolds are open subsets of , this definition of orientation preserving homeomorphisms agrees with the one defined above. With this one defines for a topological manifold with a continuous family of local orientations a maximal oriented atlas by all charts which are orientation preserving, where we orient
as above. In turn if one has a maximal oriented atlas one uses it to transport the local orientations of open subsets of
to local homological orientations of
, which are a continuous family, since the atlas is oriented.
4 Orientation of smooth manifolds
The definition of an orientation for a topological manifold needs homology groups. For smooth manifolds the definition can be simplified. To distinguish the very similar definition we call it a smooth orientation.
Definition 4.1. A smooth orientation of an -dimensional smooth manifold
is the choice of a maximal oriented atlas. Here a smooth atlas
is called oriented if the determinant of the derivatives of all coordinate changes
is positive. The oriented atlas is called maximal if it cannot be enlarged to an oriented atlas by adding another chart. Note that any oriented atlas defines a maximal oriented atlas by adding all charts such that the atlas is still oriented. This is normally the way an oriented atlas is given.
A smooth manifold is called orientable if it has a smooth orientation, otherwise it is called non-orientable.
A smooth manifold together with a smooth orientation is called an oriented smooth manifold.
5 From tangential orientations to homological orientations
For smooth manifolds we have now two definitions of an orientation, the smooth orientation and the orientation as a topological manifold. Here we explain why they are again equivalent concepts. The key observation is the following. If we have an orientation of the vector space we have defined corresponding local orientations. If we change the orientation of the vector space
, the local homological orientation changes its sign. Since there are two orientations of
as a vector space and two generators of
this correspondence is a bijection.
The next observation is that if we have a diffeomorphism from an open subset in
to another open subset
in
, then its differential preserves the standard orientation of
if and only if it preserves the corresponding local homological orientations and so the underlying homeomorphism is orientation preserving in the sense defined in the beginning of the last section.
Thus, if is smoothly oriented, i.e. is equipped with a maximal oriented smooth atlas, then - forgetting the smooth structure - we obtain an oriented topological atlas and we define the corresponding topological orientation by passing to the maximal oriented topological atlas containing these charts. In turn, if one has a maximal oriented topological atlas the subset of smooth charts in it defines a smooth orientation.
6 Reformulations of orientation for smooth manifolds
There are several equivalent formulations for orientations of smooth manifolds.
Definition 6.1. A tangential orientation of is a continuous choice of an orientation of the tangent space
in the sense of orientations of vector spaces for every point
. Here continuous means that for every
M there is a chart
around
, such that the differential of
maps for all
the orientation at
to the same orientation of
.
The relation between these two definitions is the following. If is an oriented atlas, we define an orientation of
by choosing an oriented chart
around
and define the tangential orientation as the image of the orientation of
under the differential of
. In turn, if a continuous orientation of
for all
is given, one defines a maximal oriented atlas as the atlas consisting of all charts
such that all differentials are orientation preserving, where we equip
with the induced orientation from
equipped with the standard orientation given by the canonical basis. It is easy to check that these constructions are well defined and give equivalent formulations.
Further equivalent formulations, which need a bit more knowledge of vector bundles are:
- An orientation of a smooth n-dimensional manifold is given by the reduction of the structure group
of the tangent bundle
to
, the subgroup of matrices with determinant
. That this is an equivalence is an easy exercise.
- An orientation of a smooth manifold is given by a trivialization (an isomorphism to the trivial bundle) of the exterior bundle . That this is an equivalence is an easy exercise.
Remark 6.2. Since the different concepts of orientations are all equivalent, one normally speaks of an oriented manifold in all cases. We only used the adjectives to make clear that a priory the definitions are different.
7 Criteria for orientability
There are various criteria for orientability:
Theorem 7.1. A smooth -dimensional manifold is orientable if and only if the tangent bundle (or the normal bundle of an embedding into
) has a Thom class, i.e. a class
, whose restriction to each fibre
is a generator of
. Moreover the choice of a Thom class determines an orientation and vice versa.
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![TM_x](/images/math/b/8/b/b8b036f2add61069375cfe6ac8f65851.png)
![TM](/images/math/8/e/4/8e4f2fdbb13a819d0f3d6796804f9ef4.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 7.2. A smooth manifold is orientable if and only if the first Stiefel Whitney class of its tangent bundle vanishes.
See [Milnor&Stasheff1974, Lemma 11.6 and Problem 12-A] and [Bredon1993, Proposition 17.2].
Theorem 7.3 [Dold1995, VIII Corollary 3.4]. A connected closed -dimensional manifolds
is
orientable if and only if
is non-zero, in which case it
is isomorphic to
. The choice of a generator is called a fundamental class
. The choice of
a generator corresponds to the choice of an orientation [Dold1995, VIII Definition 4.1]. For
a not necessarily connected compact oriented manifold
the components
are oriented and the sum of the fundamental classes of the components
define the fundamental class of
.
There is a generalization of Theorem 7.3 to non-compact manifolds.
Theorem 7.4 [Greenberg&Harper1981, Corollary 22.26]. If is arbitrary, then
is orientable if and only if for each
compact connected subset
there is a class
, such that for each
the map induced by the inclusion
maps
to a generator of
and the classes
mapped to each other under the maps induced by the inclusion
for all
.
The images of the classes
in
define a
homological orientation of
and in turn a homological orientation
determines the classes
.
8 Manifolds with boundary
For manifolds with boundary an orientation is defined as an orientation of its interior. An orientation of
induces an orientation on the boundary
. If
is
-dimensional, we orient the boundary, which is
-dimensional by attaching to
the local orientation
, if the restriction of a chart around
from
to
to the interior is in the oriented atlas of the interior of
. Otherwise we define
. For example it we orient the interval
by the atlas of the interior given by the identity map, then
and and
.
If the dimension of is positive, we define the induced orientation both for smooth or topological manifolds in terms of an induced maximal oriented atlas of the boundary. If
is a (smooth) chart around a boundary point
, such that its restriction to the inner is in the oriented atlas of the inner of
, then the restriction of this chart to
is a chart of
and these charts form a maximal oriented (smooth) atlas of
. The orientation given by this atlas is called the induced orientation on
.
The convention, that we consider the negative orientation on the boundary is for smooth manifolds equivalent to choosing an identification of the restriction of the tangent bundle of to
with
, where we identify
with a subbundle by selecting the outward normal vector field. With other words for smooth manifolds the induced orientation is characterized as the orientation of
, such that any outward pointing normal vector plus this orientation is the given orientation of
.
As for compact manifolds without boundary one can see that a compact connected manifold with boundary is orientable if and only if
is non-zero, in which case it is again isomorphic to
, [Dold1995]. The choice of a generator is called a relative fundamental class and again this fixes an orientation of
.
Our at the first glance slightly ad libitum looking convention is made in such a way that the following holds:
Theorem 8.1. Let be a compact oriented
-dimensional manifold with boundary. If
is the fundamental class compatible with the orientation, then
is the the fundamental class compatible with the induced orientation of the boundary as defined above.
Since the proof of this result is not in standard text books (to my knowledge), we give it here.
Proof. Since the orientation is given locally (we use the homological formulation) it is enough to show that if we consider the local orientation of in a chart near the boundary, the boundary operator maps it to the local orientation of
in the restriction of this chart to the boundary. Here we choose the chart in such a way, that the orientation of
corresponds to the standard orientation of
(if not change your atlas by a reflection in
).
Thus we consider and the local orientation given by the standard basis of
. Since we work with the half space we map the simplex constructed by the standard basis with edges
, so that it is spanned instead by
. We denote this simplex by
. The class represented by this simplex in
for some
in the inner of the simplex is the same as that of
given by the standard orientation of
. If we begin with the fundamental class
, consider its image under the boundary operator in
and pass to the local orientation at
, then it is represented by the restriction of
to the boundary of
. More precisely
, where
corresponds to the simplex spanned by the corresponding vectors, and the local orientation in
at
corresponding to the image of the fundamental class in
is given by
. But this is the negative of the local orientation of
given by the standard basis. This finishes the proof and explains why we took the negative orientation in our construction of the induced orientation in terms of an atlas.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
9 Orientation of products
Given two oriented manifolds there is an obvious way to orient their product by choosing the product atlas. If is smooth and we have given orientations as tangential orientations, we note that
is isomorphic to
and the isomorphism is induced by the differential of the projections and then the product orientation is given by the juxtaposition of the orientations of
and
.
Similarly if and
are oriented by a continuos family of local homological orientations, we note that
is isomorphic to
, this isomorphism from the latter to the first is given by the cross product
. By definition of the cross product of the local homological orientation given by the standard basis of
with the local homological orientation given by the standard basis of
is the local homological orientation given by the standard basis of
. Thus the different concepts of product orientations given by the product of an atlas and by the product of local homological orientations agree also.
As a consequence for compact oriented manifolds equipped with fundamental classes the cross product of the fundamental classes corresponds to the product of the orientations induced by the fundamental classes.
10 Orientation of complex manifold
An -dimensional complex manifold is a topological manifold together with an atlas
such that the coordinate changes are holomorphic maps. Given such an atlas the charts considered as maps to
have orientation preserving coordinate changes, since a complex matrix considered as a real matrix has determinant
, the square of the norm of the complex determinant. Thus a complex manifold considered as a real manifold has this way a canonical orientation.
11 References
- [Bredon1993] G. E. Bredon, Topology and geometry, Graduate Texts in Mathematics 139, Springer-Verlag, 1993. MR1224675 (94d:55001) Zbl 0934.55001
- [Dold1995] A. Dold, Lectures on algebraic topology, Springer-Verlag, 1995. MR1335915 (96c:55001) Zbl 0872.55001
- [Greenberg&Harper1981] M. J. Greenberg and J. R. Harper, Algebraic topology, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, 1981. MR643101 (83b:55001) Zbl 0498.55001
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
12 External links
- The Encylopedia of Mathematics article on orientation.
- The Wikipedia page on orientability.
- The Wikipedia page on orientations of vector spaces.
![\epsilon : M \to \{ \pm 1 \}](/images/math/4/6/c/46c2b13b508393e2ed97a260b9023207.png)
2 Orientation of topological manifolds
An orientation of a topological manifold is a choice of a maximal atlas, such that the coordinate changes are orientation preserving. To make this precise we have to define when a homeomorphism from an open subset of
to another open subset
is orientation preserving. We do this in terms of singular homology groups.
Definition 2.1. A homeomorphism from an open subset
of
to another open subset
is orientation preserving, if for each
the map
is the identity map. Here the isomorphisms
is the following: We first take the map to
(or to
) induced by the translation mapping
to
resp.
and then the inverse of the excision isomorphism. The isomorphism
is the inverse of the corresponding map.
Definition 2.2. An orientation of an -dimensional topological manifold
is the choice of a maximal oriented atlas. Here an atlas
is called oriented if all coordinate changes
are orientation preserving.
Note that any oriented atlas defines a maximal oriented atlas by adding all charts such that the atlas is still oriented. This is normally the way an oriented atlas is given.
A topological manifold is called orientable if it has a topological orientation, otherwise it is called non-orientable.
A topological manifold together with a topological orientation is called an oriented topological manifold.
An open subset of an oriented topological manifold is oriented by restring the charts in the maximal oriented atlas to the intersection with the open subset. A homeomorphism between oriented topological manifolds is orientation preserving if for each chart
in the oriented atlas of
the chart
is in the oriented atlas of
.
There are several equivalent formulations of orientations both for topological manifolds and for smooth manifolds which we will explain in the following sections.
3 Reformulation in terms of local homological orientations
An orientation of an -dimensional topological manifold
can also be defined in terms of the local homology groups
for each
in
.
Definition 3.1. A local homological orientation of an -dimensional topological manifold
is the choice of a generator
of the local homology group
for each
. Such a choice is called continuous, if for each
there is an open neighborhood
and a class
such that the map induced by the inclusion
maps
to
for each
. A homological orientation of
is a continuous choice of local homological orientations.
As above an open subset of
has an induced homological orientation which is given by the image under the inverse of the isomorphism induced by the inclusion
.
To get an example consider a finite dimensional oriented real vector space , i.e.
is equipped with an equivalence class of bases
, where two bases are called equivalent, if and only if the matrix of the base change matrix has positive determinant. The orientation of
as a vector space gives a homological orientation of
as a topological space as follows. We first orient at
in
by considering the simplex spanned by
. This contains
in its interior and is a generator of
. By translations we define local orientations at arbitrary points of
mapping the local orientation at
to the local orientation at
by the map induced by the translation mapping
to
. By construction this is a continuous family of local homological orientations and so gives a homological orientation of
. From this we obtain homological orientations of all open subsets of
.
The equivalence of these two concepts of an orientation of a topological manifold is shown as follows. A homeomorphism between manifolds equipped with a continuous family of local orientations is called orientation preserving if the induced map maps the corresponding local orientations to each other. We note that if both manifolds are open subsets of , this definition of orientation preserving homeomorphisms agrees with the one defined above. With this one defines for a topological manifold with a continuous family of local orientations a maximal oriented atlas by all charts which are orientation preserving, where we orient
as above. In turn if one has a maximal oriented atlas one uses it to transport the local orientations of open subsets of
to local homological orientations of
, which are a continuous family, since the atlas is oriented.
4 Orientation of smooth manifolds
The definition of an orientation for a topological manifold needs homology groups. For smooth manifolds the definition can be simplified. To distinguish the very similar definition we call it a smooth orientation.
Definition 4.1. A smooth orientation of an -dimensional smooth manifold
is the choice of a maximal oriented atlas. Here a smooth atlas
is called oriented if the determinant of the derivatives of all coordinate changes
is positive. The oriented atlas is called maximal if it cannot be enlarged to an oriented atlas by adding another chart. Note that any oriented atlas defines a maximal oriented atlas by adding all charts such that the atlas is still oriented. This is normally the way an oriented atlas is given.
A smooth manifold is called orientable if it has a smooth orientation, otherwise it is called non-orientable.
A smooth manifold together with a smooth orientation is called an oriented smooth manifold.
5 From tangential orientations to homological orientations
For smooth manifolds we have now two definitions of an orientation, the smooth orientation and the orientation as a topological manifold. Here we explain why they are again equivalent concepts. The key observation is the following. If we have an orientation of the vector space we have defined corresponding local orientations. If we change the orientation of the vector space
, the local homological orientation changes its sign. Since there are two orientations of
as a vector space and two generators of
this correspondence is a bijection.
The next observation is that if we have a diffeomorphism from an open subset in
to another open subset
in
, then its differential preserves the standard orientation of
if and only if it preserves the corresponding local homological orientations and so the underlying homeomorphism is orientation preserving in the sense defined in the beginning of the last section.
Thus, if is smoothly oriented, i.e. is equipped with a maximal oriented smooth atlas, then - forgetting the smooth structure - we obtain an oriented topological atlas and we define the corresponding topological orientation by passing to the maximal oriented topological atlas containing these charts. In turn, if one has a maximal oriented topological atlas the subset of smooth charts in it defines a smooth orientation.
6 Reformulations of orientation for smooth manifolds
There are several equivalent formulations for orientations of smooth manifolds.
Definition 6.1. A tangential orientation of is a continuous choice of an orientation of the tangent space
in the sense of orientations of vector spaces for every point
. Here continuous means that for every
M there is a chart
around
, such that the differential of
maps for all
the orientation at
to the same orientation of
.
The relation between these two definitions is the following. If is an oriented atlas, we define an orientation of
by choosing an oriented chart
around
and define the tangential orientation as the image of the orientation of
under the differential of
. In turn, if a continuous orientation of
for all
is given, one defines a maximal oriented atlas as the atlas consisting of all charts
such that all differentials are orientation preserving, where we equip
with the induced orientation from
equipped with the standard orientation given by the canonical basis. It is easy to check that these constructions are well defined and give equivalent formulations.
Further equivalent formulations, which need a bit more knowledge of vector bundles are:
- An orientation of a smooth n-dimensional manifold is given by the reduction of the structure group
of the tangent bundle
to
, the subgroup of matrices with determinant
. That this is an equivalence is an easy exercise.
- An orientation of a smooth manifold is given by a trivialization (an isomorphism to the trivial bundle) of the exterior bundle . That this is an equivalence is an easy exercise.
Remark 6.2. Since the different concepts of orientations are all equivalent, one normally speaks of an oriented manifold in all cases. We only used the adjectives to make clear that a priory the definitions are different.
7 Criteria for orientability
There are various criteria for orientability:
Theorem 7.1. A smooth -dimensional manifold is orientable if and only if the tangent bundle (or the normal bundle of an embedding into
) has a Thom class, i.e. a class
, whose restriction to each fibre
is a generator of
. Moreover the choice of a Thom class determines an orientation and vice versa.
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![TM_x](/images/math/b/8/b/b8b036f2add61069375cfe6ac8f65851.png)
![TM](/images/math/8/e/4/8e4f2fdbb13a819d0f3d6796804f9ef4.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 7.2. A smooth manifold is orientable if and only if the first Stiefel Whitney class of its tangent bundle vanishes.
See [Milnor&Stasheff1974, Lemma 11.6 and Problem 12-A] and [Bredon1993, Proposition 17.2].
Theorem 7.3 [Dold1995, VIII Corollary 3.4]. A connected closed -dimensional manifolds
is
orientable if and only if
is non-zero, in which case it
is isomorphic to
. The choice of a generator is called a fundamental class
. The choice of
a generator corresponds to the choice of an orientation [Dold1995, VIII Definition 4.1]. For
a not necessarily connected compact oriented manifold
the components
are oriented and the sum of the fundamental classes of the components
define the fundamental class of
.
There is a generalization of Theorem 7.3 to non-compact manifolds.
Theorem 7.4 [Greenberg&Harper1981, Corollary 22.26]. If is arbitrary, then
is orientable if and only if for each
compact connected subset
there is a class
, such that for each
the map induced by the inclusion
maps
to a generator of
and the classes
mapped to each other under the maps induced by the inclusion
for all
.
The images of the classes
in
define a
homological orientation of
and in turn a homological orientation
determines the classes
.
8 Manifolds with boundary
For manifolds with boundary an orientation is defined as an orientation of its interior. An orientation of
induces an orientation on the boundary
. If
is
-dimensional, we orient the boundary, which is
-dimensional by attaching to
the local orientation
, if the restriction of a chart around
from
to
to the interior is in the oriented atlas of the interior of
. Otherwise we define
. For example it we orient the interval
by the atlas of the interior given by the identity map, then
and and
.
If the dimension of is positive, we define the induced orientation both for smooth or topological manifolds in terms of an induced maximal oriented atlas of the boundary. If
is a (smooth) chart around a boundary point
, such that its restriction to the inner is in the oriented atlas of the inner of
, then the restriction of this chart to
is a chart of
and these charts form a maximal oriented (smooth) atlas of
. The orientation given by this atlas is called the induced orientation on
.
The convention, that we consider the negative orientation on the boundary is for smooth manifolds equivalent to choosing an identification of the restriction of the tangent bundle of to
with
, where we identify
with a subbundle by selecting the outward normal vector field. With other words for smooth manifolds the induced orientation is characterized as the orientation of
, such that any outward pointing normal vector plus this orientation is the given orientation of
.
As for compact manifolds without boundary one can see that a compact connected manifold with boundary is orientable if and only if
is non-zero, in which case it is again isomorphic to
, [Dold1995]. The choice of a generator is called a relative fundamental class and again this fixes an orientation of
.
Our at the first glance slightly ad libitum looking convention is made in such a way that the following holds:
Theorem 8.1. Let be a compact oriented
-dimensional manifold with boundary. If
is the fundamental class compatible with the orientation, then
is the the fundamental class compatible with the induced orientation of the boundary as defined above.
Since the proof of this result is not in standard text books (to my knowledge), we give it here.
Proof. Since the orientation is given locally (we use the homological formulation) it is enough to show that if we consider the local orientation of in a chart near the boundary, the boundary operator maps it to the local orientation of
in the restriction of this chart to the boundary. Here we choose the chart in such a way, that the orientation of
corresponds to the standard orientation of
(if not change your atlas by a reflection in
).
Thus we consider and the local orientation given by the standard basis of
. Since we work with the half space we map the simplex constructed by the standard basis with edges
, so that it is spanned instead by
. We denote this simplex by
. The class represented by this simplex in
for some
in the inner of the simplex is the same as that of
given by the standard orientation of
. If we begin with the fundamental class
, consider its image under the boundary operator in
and pass to the local orientation at
, then it is represented by the restriction of
to the boundary of
. More precisely
, where
corresponds to the simplex spanned by the corresponding vectors, and the local orientation in
at
corresponding to the image of the fundamental class in
is given by
. But this is the negative of the local orientation of
given by the standard basis. This finishes the proof and explains why we took the negative orientation in our construction of the induced orientation in terms of an atlas.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
9 Orientation of products
Given two oriented manifolds there is an obvious way to orient their product by choosing the product atlas. If is smooth and we have given orientations as tangential orientations, we note that
is isomorphic to
and the isomorphism is induced by the differential of the projections and then the product orientation is given by the juxtaposition of the orientations of
and
.
Similarly if and
are oriented by a continuos family of local homological orientations, we note that
is isomorphic to
, this isomorphism from the latter to the first is given by the cross product
. By definition of the cross product of the local homological orientation given by the standard basis of
with the local homological orientation given by the standard basis of
is the local homological orientation given by the standard basis of
. Thus the different concepts of product orientations given by the product of an atlas and by the product of local homological orientations agree also.
As a consequence for compact oriented manifolds equipped with fundamental classes the cross product of the fundamental classes corresponds to the product of the orientations induced by the fundamental classes.
10 Orientation of complex manifold
An -dimensional complex manifold is a topological manifold together with an atlas
such that the coordinate changes are holomorphic maps. Given such an atlas the charts considered as maps to
have orientation preserving coordinate changes, since a complex matrix considered as a real matrix has determinant
, the square of the norm of the complex determinant. Thus a complex manifold considered as a real manifold has this way a canonical orientation.
11 References
- [Bredon1993] G. E. Bredon, Topology and geometry, Graduate Texts in Mathematics 139, Springer-Verlag, 1993. MR1224675 (94d:55001) Zbl 0934.55001
- [Dold1995] A. Dold, Lectures on algebraic topology, Springer-Verlag, 1995. MR1335915 (96c:55001) Zbl 0872.55001
- [Greenberg&Harper1981] M. J. Greenberg and J. R. Harper, Algebraic topology, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, 1981. MR643101 (83b:55001) Zbl 0498.55001
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
12 External links
- The Encylopedia of Mathematics article on orientation.
- The Wikipedia page on orientability.
- The Wikipedia page on orientations of vector spaces.
![\epsilon : M \to \{ \pm 1 \}](/images/math/4/6/c/46c2b13b508393e2ed97a260b9023207.png)
2 Orientation of topological manifolds
An orientation of a topological manifold is a choice of a maximal atlas, such that the coordinate changes are orientation preserving. To make this precise we have to define when a homeomorphism from an open subset of
to another open subset
is orientation preserving. We do this in terms of singular homology groups.
Definition 2.1. A homeomorphism from an open subset
of
to another open subset
is orientation preserving, if for each
the map
is the identity map. Here the isomorphisms
is the following: We first take the map to
(or to
) induced by the translation mapping
to
resp.
and then the inverse of the excision isomorphism. The isomorphism
is the inverse of the corresponding map.
Definition 2.2. An orientation of an -dimensional topological manifold
is the choice of a maximal oriented atlas. Here an atlas
is called oriented if all coordinate changes
are orientation preserving.
Note that any oriented atlas defines a maximal oriented atlas by adding all charts such that the atlas is still oriented. This is normally the way an oriented atlas is given.
A topological manifold is called orientable if it has a topological orientation, otherwise it is called non-orientable.
A topological manifold together with a topological orientation is called an oriented topological manifold.
An open subset of an oriented topological manifold is oriented by restring the charts in the maximal oriented atlas to the intersection with the open subset. A homeomorphism between oriented topological manifolds is orientation preserving if for each chart
in the oriented atlas of
the chart
is in the oriented atlas of
.
There are several equivalent formulations of orientations both for topological manifolds and for smooth manifolds which we will explain in the following sections.
3 Reformulation in terms of local homological orientations
An orientation of an -dimensional topological manifold
can also be defined in terms of the local homology groups
for each
in
.
Definition 3.1. A local homological orientation of an -dimensional topological manifold
is the choice of a generator
of the local homology group
for each
. Such a choice is called continuous, if for each
there is an open neighborhood
and a class
such that the map induced by the inclusion
maps
to
for each
. A homological orientation of
is a continuous choice of local homological orientations.
As above an open subset of
has an induced homological orientation which is given by the image under the inverse of the isomorphism induced by the inclusion
.
To get an example consider a finite dimensional oriented real vector space , i.e.
is equipped with an equivalence class of bases
, where two bases are called equivalent, if and only if the matrix of the base change matrix has positive determinant. The orientation of
as a vector space gives a homological orientation of
as a topological space as follows. We first orient at
in
by considering the simplex spanned by
. This contains
in its interior and is a generator of
. By translations we define local orientations at arbitrary points of
mapping the local orientation at
to the local orientation at
by the map induced by the translation mapping
to
. By construction this is a continuous family of local homological orientations and so gives a homological orientation of
. From this we obtain homological orientations of all open subsets of
.
The equivalence of these two concepts of an orientation of a topological manifold is shown as follows. A homeomorphism between manifolds equipped with a continuous family of local orientations is called orientation preserving if the induced map maps the corresponding local orientations to each other. We note that if both manifolds are open subsets of , this definition of orientation preserving homeomorphisms agrees with the one defined above. With this one defines for a topological manifold with a continuous family of local orientations a maximal oriented atlas by all charts which are orientation preserving, where we orient
as above. In turn if one has a maximal oriented atlas one uses it to transport the local orientations of open subsets of
to local homological orientations of
, which are a continuous family, since the atlas is oriented.
4 Orientation of smooth manifolds
The definition of an orientation for a topological manifold needs homology groups. For smooth manifolds the definition can be simplified. To distinguish the very similar definition we call it a smooth orientation.
Definition 4.1. A smooth orientation of an -dimensional smooth manifold
is the choice of a maximal oriented atlas. Here a smooth atlas
is called oriented if the determinant of the derivatives of all coordinate changes
is positive. The oriented atlas is called maximal if it cannot be enlarged to an oriented atlas by adding another chart. Note that any oriented atlas defines a maximal oriented atlas by adding all charts such that the atlas is still oriented. This is normally the way an oriented atlas is given.
A smooth manifold is called orientable if it has a smooth orientation, otherwise it is called non-orientable.
A smooth manifold together with a smooth orientation is called an oriented smooth manifold.
5 From tangential orientations to homological orientations
For smooth manifolds we have now two definitions of an orientation, the smooth orientation and the orientation as a topological manifold. Here we explain why they are again equivalent concepts. The key observation is the following. If we have an orientation of the vector space we have defined corresponding local orientations. If we change the orientation of the vector space
, the local homological orientation changes its sign. Since there are two orientations of
as a vector space and two generators of
this correspondence is a bijection.
The next observation is that if we have a diffeomorphism from an open subset in
to another open subset
in
, then its differential preserves the standard orientation of
if and only if it preserves the corresponding local homological orientations and so the underlying homeomorphism is orientation preserving in the sense defined in the beginning of the last section.
Thus, if is smoothly oriented, i.e. is equipped with a maximal oriented smooth atlas, then - forgetting the smooth structure - we obtain an oriented topological atlas and we define the corresponding topological orientation by passing to the maximal oriented topological atlas containing these charts. In turn, if one has a maximal oriented topological atlas the subset of smooth charts in it defines a smooth orientation.
6 Reformulations of orientation for smooth manifolds
There are several equivalent formulations for orientations of smooth manifolds.
Definition 6.1. A tangential orientation of is a continuous choice of an orientation of the tangent space
in the sense of orientations of vector spaces for every point
. Here continuous means that for every
M there is a chart
around
, such that the differential of
maps for all
the orientation at
to the same orientation of
.
The relation between these two definitions is the following. If is an oriented atlas, we define an orientation of
by choosing an oriented chart
around
and define the tangential orientation as the image of the orientation of
under the differential of
. In turn, if a continuous orientation of
for all
is given, one defines a maximal oriented atlas as the atlas consisting of all charts
such that all differentials are orientation preserving, where we equip
with the induced orientation from
equipped with the standard orientation given by the canonical basis. It is easy to check that these constructions are well defined and give equivalent formulations.
Further equivalent formulations, which need a bit more knowledge of vector bundles are:
- An orientation of a smooth n-dimensional manifold is given by the reduction of the structure group
of the tangent bundle
to
, the subgroup of matrices with determinant
. That this is an equivalence is an easy exercise.
- An orientation of a smooth manifold is given by a trivialization (an isomorphism to the trivial bundle) of the exterior bundle . That this is an equivalence is an easy exercise.
Remark 6.2. Since the different concepts of orientations are all equivalent, one normally speaks of an oriented manifold in all cases. We only used the adjectives to make clear that a priory the definitions are different.
7 Criteria for orientability
There are various criteria for orientability:
Theorem 7.1. A smooth -dimensional manifold is orientable if and only if the tangent bundle (or the normal bundle of an embedding into
) has a Thom class, i.e. a class
, whose restriction to each fibre
is a generator of
. Moreover the choice of a Thom class determines an orientation and vice versa.
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![TM_x](/images/math/b/8/b/b8b036f2add61069375cfe6ac8f65851.png)
![TM](/images/math/8/e/4/8e4f2fdbb13a819d0f3d6796804f9ef4.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 7.2. A smooth manifold is orientable if and only if the first Stiefel Whitney class of its tangent bundle vanishes.
See [Milnor&Stasheff1974, Lemma 11.6 and Problem 12-A] and [Bredon1993, Proposition 17.2].
Theorem 7.3 [Dold1995, VIII Corollary 3.4]. A connected closed -dimensional manifolds
is
orientable if and only if
is non-zero, in which case it
is isomorphic to
. The choice of a generator is called a fundamental class
. The choice of
a generator corresponds to the choice of an orientation [Dold1995, VIII Definition 4.1]. For
a not necessarily connected compact oriented manifold
the components
are oriented and the sum of the fundamental classes of the components
define the fundamental class of
.
There is a generalization of Theorem 7.3 to non-compact manifolds.
Theorem 7.4 [Greenberg&Harper1981, Corollary 22.26]. If is arbitrary, then
is orientable if and only if for each
compact connected subset
there is a class
, such that for each
the map induced by the inclusion
maps
to a generator of
and the classes
mapped to each other under the maps induced by the inclusion
for all
.
The images of the classes
in
define a
homological orientation of
and in turn a homological orientation
determines the classes
.
8 Manifolds with boundary
For manifolds with boundary an orientation is defined as an orientation of its interior. An orientation of
induces an orientation on the boundary
. If
is
-dimensional, we orient the boundary, which is
-dimensional by attaching to
the local orientation
, if the restriction of a chart around
from
to
to the interior is in the oriented atlas of the interior of
. Otherwise we define
. For example it we orient the interval
by the atlas of the interior given by the identity map, then
and and
.
If the dimension of is positive, we define the induced orientation both for smooth or topological manifolds in terms of an induced maximal oriented atlas of the boundary. If
is a (smooth) chart around a boundary point
, such that its restriction to the inner is in the oriented atlas of the inner of
, then the restriction of this chart to
is a chart of
and these charts form a maximal oriented (smooth) atlas of
. The orientation given by this atlas is called the induced orientation on
.
The convention, that we consider the negative orientation on the boundary is for smooth manifolds equivalent to choosing an identification of the restriction of the tangent bundle of to
with
, where we identify
with a subbundle by selecting the outward normal vector field. With other words for smooth manifolds the induced orientation is characterized as the orientation of
, such that any outward pointing normal vector plus this orientation is the given orientation of
.
As for compact manifolds without boundary one can see that a compact connected manifold with boundary is orientable if and only if
is non-zero, in which case it is again isomorphic to
, [Dold1995]. The choice of a generator is called a relative fundamental class and again this fixes an orientation of
.
Our at the first glance slightly ad libitum looking convention is made in such a way that the following holds:
Theorem 8.1. Let be a compact oriented
-dimensional manifold with boundary. If
is the fundamental class compatible with the orientation, then
is the the fundamental class compatible with the induced orientation of the boundary as defined above.
Since the proof of this result is not in standard text books (to my knowledge), we give it here.
Proof. Since the orientation is given locally (we use the homological formulation) it is enough to show that if we consider the local orientation of in a chart near the boundary, the boundary operator maps it to the local orientation of
in the restriction of this chart to the boundary. Here we choose the chart in such a way, that the orientation of
corresponds to the standard orientation of
(if not change your atlas by a reflection in
).
Thus we consider and the local orientation given by the standard basis of
. Since we work with the half space we map the simplex constructed by the standard basis with edges
, so that it is spanned instead by
. We denote this simplex by
. The class represented by this simplex in
for some
in the inner of the simplex is the same as that of
given by the standard orientation of
. If we begin with the fundamental class
, consider its image under the boundary operator in
and pass to the local orientation at
, then it is represented by the restriction of
to the boundary of
. More precisely
, where
corresponds to the simplex spanned by the corresponding vectors, and the local orientation in
at
corresponding to the image of the fundamental class in
is given by
. But this is the negative of the local orientation of
given by the standard basis. This finishes the proof and explains why we took the negative orientation in our construction of the induced orientation in terms of an atlas.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
9 Orientation of products
Given two oriented manifolds there is an obvious way to orient their product by choosing the product atlas. If is smooth and we have given orientations as tangential orientations, we note that
is isomorphic to
and the isomorphism is induced by the differential of the projections and then the product orientation is given by the juxtaposition of the orientations of
and
.
Similarly if and
are oriented by a continuos family of local homological orientations, we note that
is isomorphic to
, this isomorphism from the latter to the first is given by the cross product
. By definition of the cross product of the local homological orientation given by the standard basis of
with the local homological orientation given by the standard basis of
is the local homological orientation given by the standard basis of
. Thus the different concepts of product orientations given by the product of an atlas and by the product of local homological orientations agree also.
As a consequence for compact oriented manifolds equipped with fundamental classes the cross product of the fundamental classes corresponds to the product of the orientations induced by the fundamental classes.
10 Orientation of complex manifold
An -dimensional complex manifold is a topological manifold together with an atlas
such that the coordinate changes are holomorphic maps. Given such an atlas the charts considered as maps to
have orientation preserving coordinate changes, since a complex matrix considered as a real matrix has determinant
, the square of the norm of the complex determinant. Thus a complex manifold considered as a real manifold has this way a canonical orientation.
11 References
- [Bredon1993] G. E. Bredon, Topology and geometry, Graduate Texts in Mathematics 139, Springer-Verlag, 1993. MR1224675 (94d:55001) Zbl 0934.55001
- [Dold1995] A. Dold, Lectures on algebraic topology, Springer-Verlag, 1995. MR1335915 (96c:55001) Zbl 0872.55001
- [Greenberg&Harper1981] M. J. Greenberg and J. R. Harper, Algebraic topology, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, 1981. MR643101 (83b:55001) Zbl 0498.55001
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
12 External links
- The Encylopedia of Mathematics article on orientation.
- The Wikipedia page on orientability.
- The Wikipedia page on orientations of vector spaces.
![\epsilon : M \to \{ \pm 1 \}](/images/math/4/6/c/46c2b13b508393e2ed97a260b9023207.png)
2 Orientation of topological manifolds
An orientation of a topological manifold is a choice of a maximal atlas, such that the coordinate changes are orientation preserving. To make this precise we have to define when a homeomorphism from an open subset of
to another open subset
is orientation preserving. We do this in terms of singular homology groups.
Definition 2.1. A homeomorphism from an open subset
of
to another open subset
is orientation preserving, if for each
the map
is the identity map. Here the isomorphisms
is the following: We first take the map to
(or to
) induced by the translation mapping
to
resp.
and then the inverse of the excision isomorphism. The isomorphism
is the inverse of the corresponding map.
Definition 2.2. An orientation of an -dimensional topological manifold
is the choice of a maximal oriented atlas. Here an atlas
is called oriented if all coordinate changes
are orientation preserving.
Note that any oriented atlas defines a maximal oriented atlas by adding all charts such that the atlas is still oriented. This is normally the way an oriented atlas is given.
A topological manifold is called orientable if it has a topological orientation, otherwise it is called non-orientable.
A topological manifold together with a topological orientation is called an oriented topological manifold.
An open subset of an oriented topological manifold is oriented by restring the charts in the maximal oriented atlas to the intersection with the open subset. A homeomorphism between oriented topological manifolds is orientation preserving if for each chart
in the oriented atlas of
the chart
is in the oriented atlas of
.
There are several equivalent formulations of orientations both for topological manifolds and for smooth manifolds which we will explain in the following sections.
3 Reformulation in terms of local homological orientations
An orientation of an -dimensional topological manifold
can also be defined in terms of the local homology groups
for each
in
.
Definition 3.1. A local homological orientation of an -dimensional topological manifold
is the choice of a generator
of the local homology group
for each
. Such a choice is called continuous, if for each
there is an open neighborhood
and a class
such that the map induced by the inclusion
maps
to
for each
. A homological orientation of
is a continuous choice of local homological orientations.
As above an open subset of
has an induced homological orientation which is given by the image under the inverse of the isomorphism induced by the inclusion
.
To get an example consider a finite dimensional oriented real vector space , i.e.
is equipped with an equivalence class of bases
, where two bases are called equivalent, if and only if the matrix of the base change matrix has positive determinant. The orientation of
as a vector space gives a homological orientation of
as a topological space as follows. We first orient at
in
by considering the simplex spanned by
. This contains
in its interior and is a generator of
. By translations we define local orientations at arbitrary points of
mapping the local orientation at
to the local orientation at
by the map induced by the translation mapping
to
. By construction this is a continuous family of local homological orientations and so gives a homological orientation of
. From this we obtain homological orientations of all open subsets of
.
The equivalence of these two concepts of an orientation of a topological manifold is shown as follows. A homeomorphism between manifolds equipped with a continuous family of local orientations is called orientation preserving if the induced map maps the corresponding local orientations to each other. We note that if both manifolds are open subsets of , this definition of orientation preserving homeomorphisms agrees with the one defined above. With this one defines for a topological manifold with a continuous family of local orientations a maximal oriented atlas by all charts which are orientation preserving, where we orient
as above. In turn if one has a maximal oriented atlas one uses it to transport the local orientations of open subsets of
to local homological orientations of
, which are a continuous family, since the atlas is oriented.
4 Orientation of smooth manifolds
The definition of an orientation for a topological manifold needs homology groups. For smooth manifolds the definition can be simplified. To distinguish the very similar definition we call it a smooth orientation.
Definition 4.1. A smooth orientation of an -dimensional smooth manifold
is the choice of a maximal oriented atlas. Here a smooth atlas
is called oriented if the determinant of the derivatives of all coordinate changes
is positive. The oriented atlas is called maximal if it cannot be enlarged to an oriented atlas by adding another chart. Note that any oriented atlas defines a maximal oriented atlas by adding all charts such that the atlas is still oriented. This is normally the way an oriented atlas is given.
A smooth manifold is called orientable if it has a smooth orientation, otherwise it is called non-orientable.
A smooth manifold together with a smooth orientation is called an oriented smooth manifold.
5 From tangential orientations to homological orientations
For smooth manifolds we have now two definitions of an orientation, the smooth orientation and the orientation as a topological manifold. Here we explain why they are again equivalent concepts. The key observation is the following. If we have an orientation of the vector space we have defined corresponding local orientations. If we change the orientation of the vector space
, the local homological orientation changes its sign. Since there are two orientations of
as a vector space and two generators of
this correspondence is a bijection.
The next observation is that if we have a diffeomorphism from an open subset in
to another open subset
in
, then its differential preserves the standard orientation of
if and only if it preserves the corresponding local homological orientations and so the underlying homeomorphism is orientation preserving in the sense defined in the beginning of the last section.
Thus, if is smoothly oriented, i.e. is equipped with a maximal oriented smooth atlas, then - forgetting the smooth structure - we obtain an oriented topological atlas and we define the corresponding topological orientation by passing to the maximal oriented topological atlas containing these charts. In turn, if one has a maximal oriented topological atlas the subset of smooth charts in it defines a smooth orientation.
6 Reformulations of orientation for smooth manifolds
There are several equivalent formulations for orientations of smooth manifolds.
Definition 6.1. A tangential orientation of is a continuous choice of an orientation of the tangent space
in the sense of orientations of vector spaces for every point
. Here continuous means that for every
M there is a chart
around
, such that the differential of
maps for all
the orientation at
to the same orientation of
.
The relation between these two definitions is the following. If is an oriented atlas, we define an orientation of
by choosing an oriented chart
around
and define the tangential orientation as the image of the orientation of
under the differential of
. In turn, if a continuous orientation of
for all
is given, one defines a maximal oriented atlas as the atlas consisting of all charts
such that all differentials are orientation preserving, where we equip
with the induced orientation from
equipped with the standard orientation given by the canonical basis. It is easy to check that these constructions are well defined and give equivalent formulations.
Further equivalent formulations, which need a bit more knowledge of vector bundles are:
- An orientation of a smooth n-dimensional manifold is given by the reduction of the structure group
of the tangent bundle
to
, the subgroup of matrices with determinant
. That this is an equivalence is an easy exercise.
- An orientation of a smooth manifold is given by a trivialization (an isomorphism to the trivial bundle) of the exterior bundle . That this is an equivalence is an easy exercise.
Remark 6.2. Since the different concepts of orientations are all equivalent, one normally speaks of an oriented manifold in all cases. We only used the adjectives to make clear that a priory the definitions are different.
7 Criteria for orientability
There are various criteria for orientability:
Theorem 7.1. A smooth -dimensional manifold is orientable if and only if the tangent bundle (or the normal bundle of an embedding into
) has a Thom class, i.e. a class
, whose restriction to each fibre
is a generator of
. Moreover the choice of a Thom class determines an orientation and vice versa.
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![TM_x](/images/math/b/8/b/b8b036f2add61069375cfe6ac8f65851.png)
![TM](/images/math/8/e/4/8e4f2fdbb13a819d0f3d6796804f9ef4.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 7.2. A smooth manifold is orientable if and only if the first Stiefel Whitney class of its tangent bundle vanishes.
See [Milnor&Stasheff1974, Lemma 11.6 and Problem 12-A] and [Bredon1993, Proposition 17.2].
Theorem 7.3 [Dold1995, VIII Corollary 3.4]. A connected closed -dimensional manifolds
is
orientable if and only if
is non-zero, in which case it
is isomorphic to
. The choice of a generator is called a fundamental class
. The choice of
a generator corresponds to the choice of an orientation [Dold1995, VIII Definition 4.1]. For
a not necessarily connected compact oriented manifold
the components
are oriented and the sum of the fundamental classes of the components
define the fundamental class of
.
There is a generalization of Theorem 7.3 to non-compact manifolds.
Theorem 7.4 [Greenberg&Harper1981, Corollary 22.26]. If is arbitrary, then
is orientable if and only if for each
compact connected subset
there is a class
, such that for each
the map induced by the inclusion
maps
to a generator of
and the classes
mapped to each other under the maps induced by the inclusion
for all
.
The images of the classes
in
define a
homological orientation of
and in turn a homological orientation
determines the classes
.
8 Manifolds with boundary
For manifolds with boundary an orientation is defined as an orientation of its interior. An orientation of
induces an orientation on the boundary
. If
is
-dimensional, we orient the boundary, which is
-dimensional by attaching to
the local orientation
, if the restriction of a chart around
from
to
to the interior is in the oriented atlas of the interior of
. Otherwise we define
. For example it we orient the interval
by the atlas of the interior given by the identity map, then
and and
.
If the dimension of is positive, we define the induced orientation both for smooth or topological manifolds in terms of an induced maximal oriented atlas of the boundary. If
is a (smooth) chart around a boundary point
, such that its restriction to the inner is in the oriented atlas of the inner of
, then the restriction of this chart to
is a chart of
and these charts form a maximal oriented (smooth) atlas of
. The orientation given by this atlas is called the induced orientation on
.
The convention, that we consider the negative orientation on the boundary is for smooth manifolds equivalent to choosing an identification of the restriction of the tangent bundle of to
with
, where we identify
with a subbundle by selecting the outward normal vector field. With other words for smooth manifolds the induced orientation is characterized as the orientation of
, such that any outward pointing normal vector plus this orientation is the given orientation of
.
As for compact manifolds without boundary one can see that a compact connected manifold with boundary is orientable if and only if
is non-zero, in which case it is again isomorphic to
, [Dold1995]. The choice of a generator is called a relative fundamental class and again this fixes an orientation of
.
Our at the first glance slightly ad libitum looking convention is made in such a way that the following holds:
Theorem 8.1. Let be a compact oriented
-dimensional manifold with boundary. If
is the fundamental class compatible with the orientation, then
is the the fundamental class compatible with the induced orientation of the boundary as defined above.
Since the proof of this result is not in standard text books (to my knowledge), we give it here.
Proof. Since the orientation is given locally (we use the homological formulation) it is enough to show that if we consider the local orientation of in a chart near the boundary, the boundary operator maps it to the local orientation of
in the restriction of this chart to the boundary. Here we choose the chart in such a way, that the orientation of
corresponds to the standard orientation of
(if not change your atlas by a reflection in
).
Thus we consider and the local orientation given by the standard basis of
. Since we work with the half space we map the simplex constructed by the standard basis with edges
, so that it is spanned instead by
. We denote this simplex by
. The class represented by this simplex in
for some
in the inner of the simplex is the same as that of
given by the standard orientation of
. If we begin with the fundamental class
, consider its image under the boundary operator in
and pass to the local orientation at
, then it is represented by the restriction of
to the boundary of
. More precisely
, where
corresponds to the simplex spanned by the corresponding vectors, and the local orientation in
at
corresponding to the image of the fundamental class in
is given by
. But this is the negative of the local orientation of
given by the standard basis. This finishes the proof and explains why we took the negative orientation in our construction of the induced orientation in terms of an atlas.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
9 Orientation of products
Given two oriented manifolds there is an obvious way to orient their product by choosing the product atlas. If is smooth and we have given orientations as tangential orientations, we note that
is isomorphic to
and the isomorphism is induced by the differential of the projections and then the product orientation is given by the juxtaposition of the orientations of
and
.
Similarly if and
are oriented by a continuos family of local homological orientations, we note that
is isomorphic to
, this isomorphism from the latter to the first is given by the cross product
. By definition of the cross product of the local homological orientation given by the standard basis of
with the local homological orientation given by the standard basis of
is the local homological orientation given by the standard basis of
. Thus the different concepts of product orientations given by the product of an atlas and by the product of local homological orientations agree also.
As a consequence for compact oriented manifolds equipped with fundamental classes the cross product of the fundamental classes corresponds to the product of the orientations induced by the fundamental classes.
10 Orientation of complex manifold
An -dimensional complex manifold is a topological manifold together with an atlas
such that the coordinate changes are holomorphic maps. Given such an atlas the charts considered as maps to
have orientation preserving coordinate changes, since a complex matrix considered as a real matrix has determinant
, the square of the norm of the complex determinant. Thus a complex manifold considered as a real manifold has this way a canonical orientation.
11 References
- [Bredon1993] G. E. Bredon, Topology and geometry, Graduate Texts in Mathematics 139, Springer-Verlag, 1993. MR1224675 (94d:55001) Zbl 0934.55001
- [Dold1995] A. Dold, Lectures on algebraic topology, Springer-Verlag, 1995. MR1335915 (96c:55001) Zbl 0872.55001
- [Greenberg&Harper1981] M. J. Greenberg and J. R. Harper, Algebraic topology, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, 1981. MR643101 (83b:55001) Zbl 0498.55001
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
12 External links
- The Encylopedia of Mathematics article on orientation.
- The Wikipedia page on orientability.
- The Wikipedia page on orientations of vector spaces.
![\epsilon : M \to \{ \pm 1 \}](/images/math/4/6/c/46c2b13b508393e2ed97a260b9023207.png)
2 Orientation of topological manifolds
An orientation of a topological manifold is a choice of a maximal atlas, such that the coordinate changes are orientation preserving. To make this precise we have to define when a homeomorphism from an open subset of
to another open subset
is orientation preserving. We do this in terms of singular homology groups.
Definition 2.1. A homeomorphism from an open subset
of
to another open subset
is orientation preserving, if for each
the map
is the identity map. Here the isomorphisms
is the following: We first take the map to
(or to
) induced by the translation mapping
to
resp.
and then the inverse of the excision isomorphism. The isomorphism
is the inverse of the corresponding map.
Definition 2.2. An orientation of an -dimensional topological manifold
is the choice of a maximal oriented atlas. Here an atlas
is called oriented if all coordinate changes
are orientation preserving.
Note that any oriented atlas defines a maximal oriented atlas by adding all charts such that the atlas is still oriented. This is normally the way an oriented atlas is given.
A topological manifold is called orientable if it has a topological orientation, otherwise it is called non-orientable.
A topological manifold together with a topological orientation is called an oriented topological manifold.
An open subset of an oriented topological manifold is oriented by restring the charts in the maximal oriented atlas to the intersection with the open subset. A homeomorphism between oriented topological manifolds is orientation preserving if for each chart
in the oriented atlas of
the chart
is in the oriented atlas of
.
There are several equivalent formulations of orientations both for topological manifolds and for smooth manifolds which we will explain in the following sections.
3 Reformulation in terms of local homological orientations
An orientation of an -dimensional topological manifold
can also be defined in terms of the local homology groups
for each
in
.
Definition 3.1. A local homological orientation of an -dimensional topological manifold
is the choice of a generator
of the local homology group
for each
. Such a choice is called continuous, if for each
there is an open neighborhood
and a class
such that the map induced by the inclusion
maps
to
for each
. A homological orientation of
is a continuous choice of local homological orientations.
As above an open subset of
has an induced homological orientation which is given by the image under the inverse of the isomorphism induced by the inclusion
.
To get an example consider a finite dimensional oriented real vector space , i.e.
is equipped with an equivalence class of bases
, where two bases are called equivalent, if and only if the matrix of the base change matrix has positive determinant. The orientation of
as a vector space gives a homological orientation of
as a topological space as follows. We first orient at
in
by considering the simplex spanned by
. This contains
in its interior and is a generator of
. By translations we define local orientations at arbitrary points of
mapping the local orientation at
to the local orientation at
by the map induced by the translation mapping
to
. By construction this is a continuous family of local homological orientations and so gives a homological orientation of
. From this we obtain homological orientations of all open subsets of
.
The equivalence of these two concepts of an orientation of a topological manifold is shown as follows. A homeomorphism between manifolds equipped with a continuous family of local orientations is called orientation preserving if the induced map maps the corresponding local orientations to each other. We note that if both manifolds are open subsets of , this definition of orientation preserving homeomorphisms agrees with the one defined above. With this one defines for a topological manifold with a continuous family of local orientations a maximal oriented atlas by all charts which are orientation preserving, where we orient
as above. In turn if one has a maximal oriented atlas one uses it to transport the local orientations of open subsets of
to local homological orientations of
, which are a continuous family, since the atlas is oriented.
4 Orientation of smooth manifolds
The definition of an orientation for a topological manifold needs homology groups. For smooth manifolds the definition can be simplified. To distinguish the very similar definition we call it a smooth orientation.
Definition 4.1. A smooth orientation of an -dimensional smooth manifold
is the choice of a maximal oriented atlas. Here a smooth atlas
is called oriented if the determinant of the derivatives of all coordinate changes
is positive. The oriented atlas is called maximal if it cannot be enlarged to an oriented atlas by adding another chart. Note that any oriented atlas defines a maximal oriented atlas by adding all charts such that the atlas is still oriented. This is normally the way an oriented atlas is given.
A smooth manifold is called orientable if it has a smooth orientation, otherwise it is called non-orientable.
A smooth manifold together with a smooth orientation is called an oriented smooth manifold.
5 From tangential orientations to homological orientations
For smooth manifolds we have now two definitions of an orientation, the smooth orientation and the orientation as a topological manifold. Here we explain why they are again equivalent concepts. The key observation is the following. If we have an orientation of the vector space we have defined corresponding local orientations. If we change the orientation of the vector space
, the local homological orientation changes its sign. Since there are two orientations of
as a vector space and two generators of
this correspondence is a bijection.
The next observation is that if we have a diffeomorphism from an open subset in
to another open subset
in
, then its differential preserves the standard orientation of
if and only if it preserves the corresponding local homological orientations and so the underlying homeomorphism is orientation preserving in the sense defined in the beginning of the last section.
Thus, if is smoothly oriented, i.e. is equipped with a maximal oriented smooth atlas, then - forgetting the smooth structure - we obtain an oriented topological atlas and we define the corresponding topological orientation by passing to the maximal oriented topological atlas containing these charts. In turn, if one has a maximal oriented topological atlas the subset of smooth charts in it defines a smooth orientation.
6 Reformulations of orientation for smooth manifolds
There are several equivalent formulations for orientations of smooth manifolds.
Definition 6.1. A tangential orientation of is a continuous choice of an orientation of the tangent space
in the sense of orientations of vector spaces for every point
. Here continuous means that for every
M there is a chart
around
, such that the differential of
maps for all
the orientation at
to the same orientation of
.
The relation between these two definitions is the following. If is an oriented atlas, we define an orientation of
by choosing an oriented chart
around
and define the tangential orientation as the image of the orientation of
under the differential of
. In turn, if a continuous orientation of
for all
is given, one defines a maximal oriented atlas as the atlas consisting of all charts
such that all differentials are orientation preserving, where we equip
with the induced orientation from
equipped with the standard orientation given by the canonical basis. It is easy to check that these constructions are well defined and give equivalent formulations.
Further equivalent formulations, which need a bit more knowledge of vector bundles are:
- An orientation of a smooth n-dimensional manifold is given by the reduction of the structure group
of the tangent bundle
to
, the subgroup of matrices with determinant
. That this is an equivalence is an easy exercise.
- An orientation of a smooth manifold is given by a trivialization (an isomorphism to the trivial bundle) of the exterior bundle . That this is an equivalence is an easy exercise.
Remark 6.2. Since the different concepts of orientations are all equivalent, one normally speaks of an oriented manifold in all cases. We only used the adjectives to make clear that a priory the definitions are different.
7 Criteria for orientability
There are various criteria for orientability:
Theorem 7.1. A smooth -dimensional manifold is orientable if and only if the tangent bundle (or the normal bundle of an embedding into
) has a Thom class, i.e. a class
, whose restriction to each fibre
is a generator of
. Moreover the choice of a Thom class determines an orientation and vice versa.
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![TM_x](/images/math/b/8/b/b8b036f2add61069375cfe6ac8f65851.png)
![TM](/images/math/8/e/4/8e4f2fdbb13a819d0f3d6796804f9ef4.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 7.2. A smooth manifold is orientable if and only if the first Stiefel Whitney class of its tangent bundle vanishes.
See [Milnor&Stasheff1974, Lemma 11.6 and Problem 12-A] and [Bredon1993, Proposition 17.2].
Theorem 7.3 [Dold1995, VIII Corollary 3.4]. A connected closed -dimensional manifolds
is
orientable if and only if
is non-zero, in which case it
is isomorphic to
. The choice of a generator is called a fundamental class
. The choice of
a generator corresponds to the choice of an orientation [Dold1995, VIII Definition 4.1]. For
a not necessarily connected compact oriented manifold
the components
are oriented and the sum of the fundamental classes of the components
define the fundamental class of
.
There is a generalization of Theorem 7.3 to non-compact manifolds.
Theorem 7.4 [Greenberg&Harper1981, Corollary 22.26]. If is arbitrary, then
is orientable if and only if for each
compact connected subset
there is a class
, such that for each
the map induced by the inclusion
maps
to a generator of
and the classes
mapped to each other under the maps induced by the inclusion
for all
.
The images of the classes
in
define a
homological orientation of
and in turn a homological orientation
determines the classes
.
8 Manifolds with boundary
For manifolds with boundary an orientation is defined as an orientation of its interior. An orientation of
induces an orientation on the boundary
. If
is
-dimensional, we orient the boundary, which is
-dimensional by attaching to
the local orientation
, if the restriction of a chart around
from
to
to the interior is in the oriented atlas of the interior of
. Otherwise we define
. For example it we orient the interval
by the atlas of the interior given by the identity map, then
and and
.
If the dimension of is positive, we define the induced orientation both for smooth or topological manifolds in terms of an induced maximal oriented atlas of the boundary. If
is a (smooth) chart around a boundary point
, such that its restriction to the inner is in the oriented atlas of the inner of
, then the restriction of this chart to
is a chart of
and these charts form a maximal oriented (smooth) atlas of
. The orientation given by this atlas is called the induced orientation on
.
The convention, that we consider the negative orientation on the boundary is for smooth manifolds equivalent to choosing an identification of the restriction of the tangent bundle of to
with
, where we identify
with a subbundle by selecting the outward normal vector field. With other words for smooth manifolds the induced orientation is characterized as the orientation of
, such that any outward pointing normal vector plus this orientation is the given orientation of
.
As for compact manifolds without boundary one can see that a compact connected manifold with boundary is orientable if and only if
is non-zero, in which case it is again isomorphic to
, [Dold1995]. The choice of a generator is called a relative fundamental class and again this fixes an orientation of
.
Our at the first glance slightly ad libitum looking convention is made in such a way that the following holds:
Theorem 8.1. Let be a compact oriented
-dimensional manifold with boundary. If
is the fundamental class compatible with the orientation, then
is the the fundamental class compatible with the induced orientation of the boundary as defined above.
Since the proof of this result is not in standard text books (to my knowledge), we give it here.
Proof. Since the orientation is given locally (we use the homological formulation) it is enough to show that if we consider the local orientation of in a chart near the boundary, the boundary operator maps it to the local orientation of
in the restriction of this chart to the boundary. Here we choose the chart in such a way, that the orientation of
corresponds to the standard orientation of
(if not change your atlas by a reflection in
).
Thus we consider and the local orientation given by the standard basis of
. Since we work with the half space we map the simplex constructed by the standard basis with edges
, so that it is spanned instead by
. We denote this simplex by
. The class represented by this simplex in
for some
in the inner of the simplex is the same as that of
given by the standard orientation of
. If we begin with the fundamental class
, consider its image under the boundary operator in
and pass to the local orientation at
, then it is represented by the restriction of
to the boundary of
. More precisely
, where
corresponds to the simplex spanned by the corresponding vectors, and the local orientation in
at
corresponding to the image of the fundamental class in
is given by
. But this is the negative of the local orientation of
given by the standard basis. This finishes the proof and explains why we took the negative orientation in our construction of the induced orientation in terms of an atlas.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
9 Orientation of products
Given two oriented manifolds there is an obvious way to orient their product by choosing the product atlas. If is smooth and we have given orientations as tangential orientations, we note that
is isomorphic to
and the isomorphism is induced by the differential of the projections and then the product orientation is given by the juxtaposition of the orientations of
and
.
Similarly if and
are oriented by a continuos family of local homological orientations, we note that
is isomorphic to
, this isomorphism from the latter to the first is given by the cross product
. By definition of the cross product of the local homological orientation given by the standard basis of
with the local homological orientation given by the standard basis of
is the local homological orientation given by the standard basis of
. Thus the different concepts of product orientations given by the product of an atlas and by the product of local homological orientations agree also.
As a consequence for compact oriented manifolds equipped with fundamental classes the cross product of the fundamental classes corresponds to the product of the orientations induced by the fundamental classes.
10 Orientation of complex manifold
An -dimensional complex manifold is a topological manifold together with an atlas
such that the coordinate changes are holomorphic maps. Given such an atlas the charts considered as maps to
have orientation preserving coordinate changes, since a complex matrix considered as a real matrix has determinant
, the square of the norm of the complex determinant. Thus a complex manifold considered as a real manifold has this way a canonical orientation.
11 References
- [Bredon1993] G. E. Bredon, Topology and geometry, Graduate Texts in Mathematics 139, Springer-Verlag, 1993. MR1224675 (94d:55001) Zbl 0934.55001
- [Dold1995] A. Dold, Lectures on algebraic topology, Springer-Verlag, 1995. MR1335915 (96c:55001) Zbl 0872.55001
- [Greenberg&Harper1981] M. J. Greenberg and J. R. Harper, Algebraic topology, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, 1981. MR643101 (83b:55001) Zbl 0498.55001
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
12 External links
- The Encylopedia of Mathematics article on orientation.
- The Wikipedia page on orientability.
- The Wikipedia page on orientations of vector spaces.
![\epsilon : M \to \{ \pm 1 \}](/images/math/4/6/c/46c2b13b508393e2ed97a260b9023207.png)
2 Orientation of topological manifolds
An orientation of a topological manifold is a choice of a maximal atlas, such that the coordinate changes are orientation preserving. To make this precise we have to define when a homeomorphism from an open subset of
to another open subset
is orientation preserving. We do this in terms of singular homology groups.
Definition 2.1. A homeomorphism from an open subset
of
to another open subset
is orientation preserving, if for each
the map
is the identity map. Here the isomorphisms
is the following: We first take the map to
(or to
) induced by the translation mapping
to
resp.
and then the inverse of the excision isomorphism. The isomorphism
is the inverse of the corresponding map.
Definition 2.2. An orientation of an -dimensional topological manifold
is the choice of a maximal oriented atlas. Here an atlas
is called oriented if all coordinate changes
are orientation preserving.
Note that any oriented atlas defines a maximal oriented atlas by adding all charts such that the atlas is still oriented. This is normally the way an oriented atlas is given.
A topological manifold is called orientable if it has a topological orientation, otherwise it is called non-orientable.
A topological manifold together with a topological orientation is called an oriented topological manifold.
An open subset of an oriented topological manifold is oriented by restring the charts in the maximal oriented atlas to the intersection with the open subset. A homeomorphism between oriented topological manifolds is orientation preserving if for each chart
in the oriented atlas of
the chart
is in the oriented atlas of
.
There are several equivalent formulations of orientations both for topological manifolds and for smooth manifolds which we will explain in the following sections.
3 Reformulation in terms of local homological orientations
An orientation of an -dimensional topological manifold
can also be defined in terms of the local homology groups
for each
in
.
Definition 3.1. A local homological orientation of an -dimensional topological manifold
is the choice of a generator
of the local homology group
for each
. Such a choice is called continuous, if for each
there is an open neighborhood
and a class
such that the map induced by the inclusion
maps
to
for each
. A homological orientation of
is a continuous choice of local homological orientations.
As above an open subset of
has an induced homological orientation which is given by the image under the inverse of the isomorphism induced by the inclusion
.
To get an example consider a finite dimensional oriented real vector space , i.e.
is equipped with an equivalence class of bases
, where two bases are called equivalent, if and only if the matrix of the base change matrix has positive determinant. The orientation of
as a vector space gives a homological orientation of
as a topological space as follows. We first orient at
in
by considering the simplex spanned by
. This contains
in its interior and is a generator of
. By translations we define local orientations at arbitrary points of
mapping the local orientation at
to the local orientation at
by the map induced by the translation mapping
to
. By construction this is a continuous family of local homological orientations and so gives a homological orientation of
. From this we obtain homological orientations of all open subsets of
.
The equivalence of these two concepts of an orientation of a topological manifold is shown as follows. A homeomorphism between manifolds equipped with a continuous family of local orientations is called orientation preserving if the induced map maps the corresponding local orientations to each other. We note that if both manifolds are open subsets of , this definition of orientation preserving homeomorphisms agrees with the one defined above. With this one defines for a topological manifold with a continuous family of local orientations a maximal oriented atlas by all charts which are orientation preserving, where we orient
as above. In turn if one has a maximal oriented atlas one uses it to transport the local orientations of open subsets of
to local homological orientations of
, which are a continuous family, since the atlas is oriented.
4 Orientation of smooth manifolds
The definition of an orientation for a topological manifold needs homology groups. For smooth manifolds the definition can be simplified. To distinguish the very similar definition we call it a smooth orientation.
Definition 4.1. A smooth orientation of an -dimensional smooth manifold
is the choice of a maximal oriented atlas. Here a smooth atlas
is called oriented if the determinant of the derivatives of all coordinate changes
is positive. The oriented atlas is called maximal if it cannot be enlarged to an oriented atlas by adding another chart. Note that any oriented atlas defines a maximal oriented atlas by adding all charts such that the atlas is still oriented. This is normally the way an oriented atlas is given.
A smooth manifold is called orientable if it has a smooth orientation, otherwise it is called non-orientable.
A smooth manifold together with a smooth orientation is called an oriented smooth manifold.
5 From tangential orientations to homological orientations
For smooth manifolds we have now two definitions of an orientation, the smooth orientation and the orientation as a topological manifold. Here we explain why they are again equivalent concepts. The key observation is the following. If we have an orientation of the vector space we have defined corresponding local orientations. If we change the orientation of the vector space
, the local homological orientation changes its sign. Since there are two orientations of
as a vector space and two generators of
this correspondence is a bijection.
The next observation is that if we have a diffeomorphism from an open subset in
to another open subset
in
, then its differential preserves the standard orientation of
if and only if it preserves the corresponding local homological orientations and so the underlying homeomorphism is orientation preserving in the sense defined in the beginning of the last section.
Thus, if is smoothly oriented, i.e. is equipped with a maximal oriented smooth atlas, then - forgetting the smooth structure - we obtain an oriented topological atlas and we define the corresponding topological orientation by passing to the maximal oriented topological atlas containing these charts. In turn, if one has a maximal oriented topological atlas the subset of smooth charts in it defines a smooth orientation.
6 Reformulations of orientation for smooth manifolds
There are several equivalent formulations for orientations of smooth manifolds.
Definition 6.1. A tangential orientation of is a continuous choice of an orientation of the tangent space
in the sense of orientations of vector spaces for every point
. Here continuous means that for every
M there is a chart
around
, such that the differential of
maps for all
the orientation at
to the same orientation of
.
The relation between these two definitions is the following. If is an oriented atlas, we define an orientation of
by choosing an oriented chart
around
and define the tangential orientation as the image of the orientation of
under the differential of
. In turn, if a continuous orientation of
for all
is given, one defines a maximal oriented atlas as the atlas consisting of all charts
such that all differentials are orientation preserving, where we equip
with the induced orientation from
equipped with the standard orientation given by the canonical basis. It is easy to check that these constructions are well defined and give equivalent formulations.
Further equivalent formulations, which need a bit more knowledge of vector bundles are:
- An orientation of a smooth n-dimensional manifold is given by the reduction of the structure group
of the tangent bundle
to
, the subgroup of matrices with determinant
. That this is an equivalence is an easy exercise.
- An orientation of a smooth manifold is given by a trivialization (an isomorphism to the trivial bundle) of the exterior bundle . That this is an equivalence is an easy exercise.
Remark 6.2. Since the different concepts of orientations are all equivalent, one normally speaks of an oriented manifold in all cases. We only used the adjectives to make clear that a priory the definitions are different.
7 Criteria for orientability
There are various criteria for orientability:
Theorem 7.1. A smooth -dimensional manifold is orientable if and only if the tangent bundle (or the normal bundle of an embedding into
) has a Thom class, i.e. a class
, whose restriction to each fibre
is a generator of
. Moreover the choice of a Thom class determines an orientation and vice versa.
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![TM_x](/images/math/b/8/b/b8b036f2add61069375cfe6ac8f65851.png)
![TM](/images/math/8/e/4/8e4f2fdbb13a819d0f3d6796804f9ef4.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 7.2. A smooth manifold is orientable if and only if the first Stiefel Whitney class of its tangent bundle vanishes.
See [Milnor&Stasheff1974, Lemma 11.6 and Problem 12-A] and [Bredon1993, Proposition 17.2].
Theorem 7.3 [Dold1995, VIII Corollary 3.4]. A connected closed -dimensional manifolds
is
orientable if and only if
is non-zero, in which case it
is isomorphic to
. The choice of a generator is called a fundamental class
. The choice of
a generator corresponds to the choice of an orientation [Dold1995, VIII Definition 4.1]. For
a not necessarily connected compact oriented manifold
the components
are oriented and the sum of the fundamental classes of the components
define the fundamental class of
.
There is a generalization of Theorem 7.3 to non-compact manifolds.
Theorem 7.4 [Greenberg&Harper1981, Corollary 22.26]. If is arbitrary, then
is orientable if and only if for each
compact connected subset
there is a class
, such that for each
the map induced by the inclusion
maps
to a generator of
and the classes
mapped to each other under the maps induced by the inclusion
for all
.
The images of the classes
in
define a
homological orientation of
and in turn a homological orientation
determines the classes
.
8 Manifolds with boundary
For manifolds with boundary an orientation is defined as an orientation of its interior. An orientation of
induces an orientation on the boundary
. If
is
-dimensional, we orient the boundary, which is
-dimensional by attaching to
the local orientation
, if the restriction of a chart around
from
to
to the interior is in the oriented atlas of the interior of
. Otherwise we define
. For example it we orient the interval
by the atlas of the interior given by the identity map, then
and and
.
If the dimension of is positive, we define the induced orientation both for smooth or topological manifolds in terms of an induced maximal oriented atlas of the boundary. If
is a (smooth) chart around a boundary point
, such that its restriction to the inner is in the oriented atlas of the inner of
, then the restriction of this chart to
is a chart of
and these charts form a maximal oriented (smooth) atlas of
. The orientation given by this atlas is called the induced orientation on
.
The convention, that we consider the negative orientation on the boundary is for smooth manifolds equivalent to choosing an identification of the restriction of the tangent bundle of to
with
, where we identify
with a subbundle by selecting the outward normal vector field. With other words for smooth manifolds the induced orientation is characterized as the orientation of
, such that any outward pointing normal vector plus this orientation is the given orientation of
.
As for compact manifolds without boundary one can see that a compact connected manifold with boundary is orientable if and only if
is non-zero, in which case it is again isomorphic to
, [Dold1995]. The choice of a generator is called a relative fundamental class and again this fixes an orientation of
.
Our at the first glance slightly ad libitum looking convention is made in such a way that the following holds:
Theorem 8.1. Let be a compact oriented
-dimensional manifold with boundary. If
is the fundamental class compatible with the orientation, then
is the the fundamental class compatible with the induced orientation of the boundary as defined above.
Since the proof of this result is not in standard text books (to my knowledge), we give it here.
Proof. Since the orientation is given locally (we use the homological formulation) it is enough to show that if we consider the local orientation of in a chart near the boundary, the boundary operator maps it to the local orientation of
in the restriction of this chart to the boundary. Here we choose the chart in such a way, that the orientation of
corresponds to the standard orientation of
(if not change your atlas by a reflection in
).
Thus we consider and the local orientation given by the standard basis of
. Since we work with the half space we map the simplex constructed by the standard basis with edges
, so that it is spanned instead by
. We denote this simplex by
. The class represented by this simplex in
for some
in the inner of the simplex is the same as that of
given by the standard orientation of
. If we begin with the fundamental class
, consider its image under the boundary operator in
and pass to the local orientation at
, then it is represented by the restriction of
to the boundary of
. More precisely
, where
corresponds to the simplex spanned by the corresponding vectors, and the local orientation in
at
corresponding to the image of the fundamental class in
is given by
. But this is the negative of the local orientation of
given by the standard basis. This finishes the proof and explains why we took the negative orientation in our construction of the induced orientation in terms of an atlas.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
9 Orientation of products
Given two oriented manifolds there is an obvious way to orient their product by choosing the product atlas. If is smooth and we have given orientations as tangential orientations, we note that
is isomorphic to
and the isomorphism is induced by the differential of the projections and then the product orientation is given by the juxtaposition of the orientations of
and
.
Similarly if and
are oriented by a continuos family of local homological orientations, we note that
is isomorphic to
, this isomorphism from the latter to the first is given by the cross product
. By definition of the cross product of the local homological orientation given by the standard basis of
with the local homological orientation given by the standard basis of
is the local homological orientation given by the standard basis of
. Thus the different concepts of product orientations given by the product of an atlas and by the product of local homological orientations agree also.
As a consequence for compact oriented manifolds equipped with fundamental classes the cross product of the fundamental classes corresponds to the product of the orientations induced by the fundamental classes.
10 Orientation of complex manifold
An -dimensional complex manifold is a topological manifold together with an atlas
such that the coordinate changes are holomorphic maps. Given such an atlas the charts considered as maps to
have orientation preserving coordinate changes, since a complex matrix considered as a real matrix has determinant
, the square of the norm of the complex determinant. Thus a complex manifold considered as a real manifold has this way a canonical orientation.
11 References
- [Bredon1993] G. E. Bredon, Topology and geometry, Graduate Texts in Mathematics 139, Springer-Verlag, 1993. MR1224675 (94d:55001) Zbl 0934.55001
- [Dold1995] A. Dold, Lectures on algebraic topology, Springer-Verlag, 1995. MR1335915 (96c:55001) Zbl 0872.55001
- [Greenberg&Harper1981] M. J. Greenberg and J. R. Harper, Algebraic topology, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, 1981. MR643101 (83b:55001) Zbl 0498.55001
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
12 External links
- The Encylopedia of Mathematics article on orientation.
- The Wikipedia page on orientability.
- The Wikipedia page on orientations of vector spaces.
![\epsilon : M \to \{ \pm 1 \}](/images/math/4/6/c/46c2b13b508393e2ed97a260b9023207.png)
2 Orientation of topological manifolds
An orientation of a topological manifold is a choice of a maximal atlas, such that the coordinate changes are orientation preserving. To make this precise we have to define when a homeomorphism from an open subset of
to another open subset
is orientation preserving. We do this in terms of singular homology groups.
Definition 2.1. A homeomorphism from an open subset
of
to another open subset
is orientation preserving, if for each
the map
is the identity map. Here the isomorphisms
is the following: We first take the map to
(or to
) induced by the translation mapping
to
resp.
and then the inverse of the excision isomorphism. The isomorphism
is the inverse of the corresponding map.
Definition 2.2. An orientation of an -dimensional topological manifold
is the choice of a maximal oriented atlas. Here an atlas
is called oriented if all coordinate changes
are orientation preserving.
Note that any oriented atlas defines a maximal oriented atlas by adding all charts such that the atlas is still oriented. This is normally the way an oriented atlas is given.
A topological manifold is called orientable if it has a topological orientation, otherwise it is called non-orientable.
A topological manifold together with a topological orientation is called an oriented topological manifold.
An open subset of an oriented topological manifold is oriented by restring the charts in the maximal oriented atlas to the intersection with the open subset. A homeomorphism between oriented topological manifolds is orientation preserving if for each chart
in the oriented atlas of
the chart
is in the oriented atlas of
.
There are several equivalent formulations of orientations both for topological manifolds and for smooth manifolds which we will explain in the following sections.
3 Reformulation in terms of local homological orientations
An orientation of an -dimensional topological manifold
can also be defined in terms of the local homology groups
for each
in
.
Definition 3.1. A local homological orientation of an -dimensional topological manifold
is the choice of a generator
of the local homology group
for each
. Such a choice is called continuous, if for each
there is an open neighborhood
and a class
such that the map induced by the inclusion
maps
to
for each
. A homological orientation of
is a continuous choice of local homological orientations.
As above an open subset of
has an induced homological orientation which is given by the image under the inverse of the isomorphism induced by the inclusion
.
To get an example consider a finite dimensional oriented real vector space , i.e.
is equipped with an equivalence class of bases
, where two bases are called equivalent, if and only if the matrix of the base change matrix has positive determinant. The orientation of
as a vector space gives a homological orientation of
as a topological space as follows. We first orient at
in
by considering the simplex spanned by
. This contains
in its interior and is a generator of
. By translations we define local orientations at arbitrary points of
mapping the local orientation at
to the local orientation at
by the map induced by the translation mapping
to
. By construction this is a continuous family of local homological orientations and so gives a homological orientation of
. From this we obtain homological orientations of all open subsets of
.
The equivalence of these two concepts of an orientation of a topological manifold is shown as follows. A homeomorphism between manifolds equipped with a continuous family of local orientations is called orientation preserving if the induced map maps the corresponding local orientations to each other. We note that if both manifolds are open subsets of , this definition of orientation preserving homeomorphisms agrees with the one defined above. With this one defines for a topological manifold with a continuous family of local orientations a maximal oriented atlas by all charts which are orientation preserving, where we orient
as above. In turn if one has a maximal oriented atlas one uses it to transport the local orientations of open subsets of
to local homological orientations of
, which are a continuous family, since the atlas is oriented.
4 Orientation of smooth manifolds
The definition of an orientation for a topological manifold needs homology groups. For smooth manifolds the definition can be simplified. To distinguish the very similar definition we call it a smooth orientation.
Definition 4.1. A smooth orientation of an -dimensional smooth manifold
is the choice of a maximal oriented atlas. Here a smooth atlas
is called oriented if the determinant of the derivatives of all coordinate changes
is positive. The oriented atlas is called maximal if it cannot be enlarged to an oriented atlas by adding another chart. Note that any oriented atlas defines a maximal oriented atlas by adding all charts such that the atlas is still oriented. This is normally the way an oriented atlas is given.
A smooth manifold is called orientable if it has a smooth orientation, otherwise it is called non-orientable.
A smooth manifold together with a smooth orientation is called an oriented smooth manifold.
5 From tangential orientations to homological orientations
For smooth manifolds we have now two definitions of an orientation, the smooth orientation and the orientation as a topological manifold. Here we explain why they are again equivalent concepts. The key observation is the following. If we have an orientation of the vector space we have defined corresponding local orientations. If we change the orientation of the vector space
, the local homological orientation changes its sign. Since there are two orientations of
as a vector space and two generators of
this correspondence is a bijection.
The next observation is that if we have a diffeomorphism from an open subset in
to another open subset
in
, then its differential preserves the standard orientation of
if and only if it preserves the corresponding local homological orientations and so the underlying homeomorphism is orientation preserving in the sense defined in the beginning of the last section.
Thus, if is smoothly oriented, i.e. is equipped with a maximal oriented smooth atlas, then - forgetting the smooth structure - we obtain an oriented topological atlas and we define the corresponding topological orientation by passing to the maximal oriented topological atlas containing these charts. In turn, if one has a maximal oriented topological atlas the subset of smooth charts in it defines a smooth orientation.
6 Reformulations of orientation for smooth manifolds
There are several equivalent formulations for orientations of smooth manifolds.
Definition 6.1. A tangential orientation of is a continuous choice of an orientation of the tangent space
in the sense of orientations of vector spaces for every point
. Here continuous means that for every
M there is a chart
around
, such that the differential of
maps for all
the orientation at
to the same orientation of
.
The relation between these two definitions is the following. If is an oriented atlas, we define an orientation of
by choosing an oriented chart
around
and define the tangential orientation as the image of the orientation of
under the differential of
. In turn, if a continuous orientation of
for all
is given, one defines a maximal oriented atlas as the atlas consisting of all charts
such that all differentials are orientation preserving, where we equip
with the induced orientation from
equipped with the standard orientation given by the canonical basis. It is easy to check that these constructions are well defined and give equivalent formulations.
Further equivalent formulations, which need a bit more knowledge of vector bundles are:
- An orientation of a smooth n-dimensional manifold is given by the reduction of the structure group
of the tangent bundle
to
, the subgroup of matrices with determinant
. That this is an equivalence is an easy exercise.
- An orientation of a smooth manifold is given by a trivialization (an isomorphism to the trivial bundle) of the exterior bundle . That this is an equivalence is an easy exercise.
Remark 6.2. Since the different concepts of orientations are all equivalent, one normally speaks of an oriented manifold in all cases. We only used the adjectives to make clear that a priory the definitions are different.
7 Criteria for orientability
There are various criteria for orientability:
Theorem 7.1. A smooth -dimensional manifold is orientable if and only if the tangent bundle (or the normal bundle of an embedding into
) has a Thom class, i.e. a class
, whose restriction to each fibre
is a generator of
. Moreover the choice of a Thom class determines an orientation and vice versa.
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![TM_x](/images/math/b/8/b/b8b036f2add61069375cfe6ac8f65851.png)
![TM](/images/math/8/e/4/8e4f2fdbb13a819d0f3d6796804f9ef4.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 7.2. A smooth manifold is orientable if and only if the first Stiefel Whitney class of its tangent bundle vanishes.
See [Milnor&Stasheff1974, Lemma 11.6 and Problem 12-A] and [Bredon1993, Proposition 17.2].
Theorem 7.3 [Dold1995, VIII Corollary 3.4]. A connected closed -dimensional manifolds
is
orientable if and only if
is non-zero, in which case it
is isomorphic to
. The choice of a generator is called a fundamental class
. The choice of
a generator corresponds to the choice of an orientation [Dold1995, VIII Definition 4.1]. For
a not necessarily connected compact oriented manifold
the components
are oriented and the sum of the fundamental classes of the components
define the fundamental class of
.
There is a generalization of Theorem 7.3 to non-compact manifolds.
Theorem 7.4 [Greenberg&Harper1981, Corollary 22.26]. If is arbitrary, then
is orientable if and only if for each
compact connected subset
there is a class
, such that for each
the map induced by the inclusion
maps
to a generator of
and the classes
mapped to each other under the maps induced by the inclusion
for all
.
The images of the classes
in
define a
homological orientation of
and in turn a homological orientation
determines the classes
.
8 Manifolds with boundary
For manifolds with boundary an orientation is defined as an orientation of its interior. An orientation of
induces an orientation on the boundary
. If
is
-dimensional, we orient the boundary, which is
-dimensional by attaching to
the local orientation
, if the restriction of a chart around
from
to
to the interior is in the oriented atlas of the interior of
. Otherwise we define
. For example it we orient the interval
by the atlas of the interior given by the identity map, then
and and
.
If the dimension of is positive, we define the induced orientation both for smooth or topological manifolds in terms of an induced maximal oriented atlas of the boundary. If
is a (smooth) chart around a boundary point
, such that its restriction to the inner is in the oriented atlas of the inner of
, then the restriction of this chart to
is a chart of
and these charts form a maximal oriented (smooth) atlas of
. The orientation given by this atlas is called the induced orientation on
.
The convention, that we consider the negative orientation on the boundary is for smooth manifolds equivalent to choosing an identification of the restriction of the tangent bundle of to
with
, where we identify
with a subbundle by selecting the outward normal vector field. With other words for smooth manifolds the induced orientation is characterized as the orientation of
, such that any outward pointing normal vector plus this orientation is the given orientation of
.
As for compact manifolds without boundary one can see that a compact connected manifold with boundary is orientable if and only if
is non-zero, in which case it is again isomorphic to
, [Dold1995]. The choice of a generator is called a relative fundamental class and again this fixes an orientation of
.
Our at the first glance slightly ad libitum looking convention is made in such a way that the following holds:
Theorem 8.1. Let be a compact oriented
-dimensional manifold with boundary. If
is the fundamental class compatible with the orientation, then
is the the fundamental class compatible with the induced orientation of the boundary as defined above.
Since the proof of this result is not in standard text books (to my knowledge), we give it here.
Proof. Since the orientation is given locally (we use the homological formulation) it is enough to show that if we consider the local orientation of in a chart near the boundary, the boundary operator maps it to the local orientation of
in the restriction of this chart to the boundary. Here we choose the chart in such a way, that the orientation of
corresponds to the standard orientation of
(if not change your atlas by a reflection in
).
Thus we consider and the local orientation given by the standard basis of
. Since we work with the half space we map the simplex constructed by the standard basis with edges
, so that it is spanned instead by
. We denote this simplex by
. The class represented by this simplex in
for some
in the inner of the simplex is the same as that of
given by the standard orientation of
. If we begin with the fundamental class
, consider its image under the boundary operator in
and pass to the local orientation at
, then it is represented by the restriction of
to the boundary of
. More precisely
, where
corresponds to the simplex spanned by the corresponding vectors, and the local orientation in
at
corresponding to the image of the fundamental class in
is given by
. But this is the negative of the local orientation of
given by the standard basis. This finishes the proof and explains why we took the negative orientation in our construction of the induced orientation in terms of an atlas.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
9 Orientation of products
Given two oriented manifolds there is an obvious way to orient their product by choosing the product atlas. If is smooth and we have given orientations as tangential orientations, we note that
is isomorphic to
and the isomorphism is induced by the differential of the projections and then the product orientation is given by the juxtaposition of the orientations of
and
.
Similarly if and
are oriented by a continuos family of local homological orientations, we note that
is isomorphic to
, this isomorphism from the latter to the first is given by the cross product
. By definition of the cross product of the local homological orientation given by the standard basis of
with the local homological orientation given by the standard basis of
is the local homological orientation given by the standard basis of
. Thus the different concepts of product orientations given by the product of an atlas and by the product of local homological orientations agree also.
As a consequence for compact oriented manifolds equipped with fundamental classes the cross product of the fundamental classes corresponds to the product of the orientations induced by the fundamental classes.
10 Orientation of complex manifold
An -dimensional complex manifold is a topological manifold together with an atlas
such that the coordinate changes are holomorphic maps. Given such an atlas the charts considered as maps to
have orientation preserving coordinate changes, since a complex matrix considered as a real matrix has determinant
, the square of the norm of the complex determinant. Thus a complex manifold considered as a real manifold has this way a canonical orientation.
11 References
- [Bredon1993] G. E. Bredon, Topology and geometry, Graduate Texts in Mathematics 139, Springer-Verlag, 1993. MR1224675 (94d:55001) Zbl 0934.55001
- [Dold1995] A. Dold, Lectures on algebraic topology, Springer-Verlag, 1995. MR1335915 (96c:55001) Zbl 0872.55001
- [Greenberg&Harper1981] M. J. Greenberg and J. R. Harper, Algebraic topology, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, 1981. MR643101 (83b:55001) Zbl 0498.55001
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
12 External links
- The Encylopedia of Mathematics article on orientation.
- The Wikipedia page on orientability.
- The Wikipedia page on orientations of vector spaces.
![\epsilon : M \to \{ \pm 1 \}](/images/math/4/6/c/46c2b13b508393e2ed97a260b9023207.png)
2 Orientation of topological manifolds
An orientation of a topological manifold is a choice of a maximal atlas, such that the coordinate changes are orientation preserving. To make this precise we have to define when a homeomorphism from an open subset of
to another open subset
is orientation preserving. We do this in terms of singular homology groups.
Definition 2.1. A homeomorphism from an open subset
of
to another open subset
is orientation preserving, if for each
the map
is the identity map. Here the isomorphisms
is the following: We first take the map to
(or to
) induced by the translation mapping
to
resp.
and then the inverse of the excision isomorphism. The isomorphism
is the inverse of the corresponding map.
Definition 2.2. An orientation of an -dimensional topological manifold
is the choice of a maximal oriented atlas. Here an atlas
is called oriented if all coordinate changes
are orientation preserving.
Note that any oriented atlas defines a maximal oriented atlas by adding all charts such that the atlas is still oriented. This is normally the way an oriented atlas is given.
A topological manifold is called orientable if it has a topological orientation, otherwise it is called non-orientable.
A topological manifold together with a topological orientation is called an oriented topological manifold.
An open subset of an oriented topological manifold is oriented by restring the charts in the maximal oriented atlas to the intersection with the open subset. A homeomorphism between oriented topological manifolds is orientation preserving if for each chart
in the oriented atlas of
the chart
is in the oriented atlas of
.
There are several equivalent formulations of orientations both for topological manifolds and for smooth manifolds which we will explain in the following sections.
3 Reformulation in terms of local homological orientations
An orientation of an -dimensional topological manifold
can also be defined in terms of the local homology groups
for each
in
.
Definition 3.1. A local homological orientation of an -dimensional topological manifold
is the choice of a generator
of the local homology group
for each
. Such a choice is called continuous, if for each
there is an open neighborhood
and a class
such that the map induced by the inclusion
maps
to
for each
. A homological orientation of
is a continuous choice of local homological orientations.
As above an open subset of
has an induced homological orientation which is given by the image under the inverse of the isomorphism induced by the inclusion
.
To get an example consider a finite dimensional oriented real vector space , i.e.
is equipped with an equivalence class of bases
, where two bases are called equivalent, if and only if the matrix of the base change matrix has positive determinant. The orientation of
as a vector space gives a homological orientation of
as a topological space as follows. We first orient at
in
by considering the simplex spanned by
. This contains
in its interior and is a generator of
. By translations we define local orientations at arbitrary points of
mapping the local orientation at
to the local orientation at
by the map induced by the translation mapping
to
. By construction this is a continuous family of local homological orientations and so gives a homological orientation of
. From this we obtain homological orientations of all open subsets of
.
The equivalence of these two concepts of an orientation of a topological manifold is shown as follows. A homeomorphism between manifolds equipped with a continuous family of local orientations is called orientation preserving if the induced map maps the corresponding local orientations to each other. We note that if both manifolds are open subsets of , this definition of orientation preserving homeomorphisms agrees with the one defined above. With this one defines for a topological manifold with a continuous family of local orientations a maximal oriented atlas by all charts which are orientation preserving, where we orient
as above. In turn if one has a maximal oriented atlas one uses it to transport the local orientations of open subsets of
to local homological orientations of
, which are a continuous family, since the atlas is oriented.
4 Orientation of smooth manifolds
The definition of an orientation for a topological manifold needs homology groups. For smooth manifolds the definition can be simplified. To distinguish the very similar definition we call it a smooth orientation.
Definition 4.1. A smooth orientation of an -dimensional smooth manifold
is the choice of a maximal oriented atlas. Here a smooth atlas
is called oriented if the determinant of the derivatives of all coordinate changes
is positive. The oriented atlas is called maximal if it cannot be enlarged to an oriented atlas by adding another chart. Note that any oriented atlas defines a maximal oriented atlas by adding all charts such that the atlas is still oriented. This is normally the way an oriented atlas is given.
A smooth manifold is called orientable if it has a smooth orientation, otherwise it is called non-orientable.
A smooth manifold together with a smooth orientation is called an oriented smooth manifold.
5 From tangential orientations to homological orientations
For smooth manifolds we have now two definitions of an orientation, the smooth orientation and the orientation as a topological manifold. Here we explain why they are again equivalent concepts. The key observation is the following. If we have an orientation of the vector space we have defined corresponding local orientations. If we change the orientation of the vector space
, the local homological orientation changes its sign. Since there are two orientations of
as a vector space and two generators of
this correspondence is a bijection.
The next observation is that if we have a diffeomorphism from an open subset in
to another open subset
in
, then its differential preserves the standard orientation of
if and only if it preserves the corresponding local homological orientations and so the underlying homeomorphism is orientation preserving in the sense defined in the beginning of the last section.
Thus, if is smoothly oriented, i.e. is equipped with a maximal oriented smooth atlas, then - forgetting the smooth structure - we obtain an oriented topological atlas and we define the corresponding topological orientation by passing to the maximal oriented topological atlas containing these charts. In turn, if one has a maximal oriented topological atlas the subset of smooth charts in it defines a smooth orientation.
6 Reformulations of orientation for smooth manifolds
There are several equivalent formulations for orientations of smooth manifolds.
Definition 6.1. A tangential orientation of is a continuous choice of an orientation of the tangent space
in the sense of orientations of vector spaces for every point
. Here continuous means that for every
M there is a chart
around
, such that the differential of
maps for all
the orientation at
to the same orientation of
.
The relation between these two definitions is the following. If is an oriented atlas, we define an orientation of
by choosing an oriented chart
around
and define the tangential orientation as the image of the orientation of
under the differential of
. In turn, if a continuous orientation of
for all
is given, one defines a maximal oriented atlas as the atlas consisting of all charts
such that all differentials are orientation preserving, where we equip
with the induced orientation from
equipped with the standard orientation given by the canonical basis. It is easy to check that these constructions are well defined and give equivalent formulations.
Further equivalent formulations, which need a bit more knowledge of vector bundles are:
- An orientation of a smooth n-dimensional manifold is given by the reduction of the structure group
of the tangent bundle
to
, the subgroup of matrices with determinant
. That this is an equivalence is an easy exercise.
- An orientation of a smooth manifold is given by a trivialization (an isomorphism to the trivial bundle) of the exterior bundle . That this is an equivalence is an easy exercise.
Remark 6.2. Since the different concepts of orientations are all equivalent, one normally speaks of an oriented manifold in all cases. We only used the adjectives to make clear that a priory the definitions are different.
7 Criteria for orientability
There are various criteria for orientability:
Theorem 7.1. A smooth -dimensional manifold is orientable if and only if the tangent bundle (or the normal bundle of an embedding into
) has a Thom class, i.e. a class
, whose restriction to each fibre
is a generator of
. Moreover the choice of a Thom class determines an orientation and vice versa.
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![TM_x](/images/math/b/8/b/b8b036f2add61069375cfe6ac8f65851.png)
![TM](/images/math/8/e/4/8e4f2fdbb13a819d0f3d6796804f9ef4.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 7.2. A smooth manifold is orientable if and only if the first Stiefel Whitney class of its tangent bundle vanishes.
See [Milnor&Stasheff1974, Lemma 11.6 and Problem 12-A] and [Bredon1993, Proposition 17.2].
Theorem 7.3 [Dold1995, VIII Corollary 3.4]. A connected closed -dimensional manifolds
is
orientable if and only if
is non-zero, in which case it
is isomorphic to
. The choice of a generator is called a fundamental class
. The choice of
a generator corresponds to the choice of an orientation [Dold1995, VIII Definition 4.1]. For
a not necessarily connected compact oriented manifold
the components
are oriented and the sum of the fundamental classes of the components
define the fundamental class of
.
There is a generalization of Theorem 7.3 to non-compact manifolds.
Theorem 7.4 [Greenberg&Harper1981, Corollary 22.26]. If is arbitrary, then
is orientable if and only if for each
compact connected subset
there is a class
, such that for each
the map induced by the inclusion
maps
to a generator of
and the classes
mapped to each other under the maps induced by the inclusion
for all
.
The images of the classes
in
define a
homological orientation of
and in turn a homological orientation
determines the classes
.
8 Manifolds with boundary
For manifolds with boundary an orientation is defined as an orientation of its interior. An orientation of
induces an orientation on the boundary
. If
is
-dimensional, we orient the boundary, which is
-dimensional by attaching to
the local orientation
, if the restriction of a chart around
from
to
to the interior is in the oriented atlas of the interior of
. Otherwise we define
. For example it we orient the interval
by the atlas of the interior given by the identity map, then
and and
.
If the dimension of is positive, we define the induced orientation both for smooth or topological manifolds in terms of an induced maximal oriented atlas of the boundary. If
is a (smooth) chart around a boundary point
, such that its restriction to the inner is in the oriented atlas of the inner of
, then the restriction of this chart to
is a chart of
and these charts form a maximal oriented (smooth) atlas of
. The orientation given by this atlas is called the induced orientation on
.
The convention, that we consider the negative orientation on the boundary is for smooth manifolds equivalent to choosing an identification of the restriction of the tangent bundle of to
with
, where we identify
with a subbundle by selecting the outward normal vector field. With other words for smooth manifolds the induced orientation is characterized as the orientation of
, such that any outward pointing normal vector plus this orientation is the given orientation of
.
As for compact manifolds without boundary one can see that a compact connected manifold with boundary is orientable if and only if
is non-zero, in which case it is again isomorphic to
, [Dold1995]. The choice of a generator is called a relative fundamental class and again this fixes an orientation of
.
Our at the first glance slightly ad libitum looking convention is made in such a way that the following holds:
Theorem 8.1. Let be a compact oriented
-dimensional manifold with boundary. If
is the fundamental class compatible with the orientation, then
is the the fundamental class compatible with the induced orientation of the boundary as defined above.
Since the proof of this result is not in standard text books (to my knowledge), we give it here.
Proof. Since the orientation is given locally (we use the homological formulation) it is enough to show that if we consider the local orientation of in a chart near the boundary, the boundary operator maps it to the local orientation of
in the restriction of this chart to the boundary. Here we choose the chart in such a way, that the orientation of
corresponds to the standard orientation of
(if not change your atlas by a reflection in
).
Thus we consider and the local orientation given by the standard basis of
. Since we work with the half space we map the simplex constructed by the standard basis with edges
, so that it is spanned instead by
. We denote this simplex by
. The class represented by this simplex in
for some
in the inner of the simplex is the same as that of
given by the standard orientation of
. If we begin with the fundamental class
, consider its image under the boundary operator in
and pass to the local orientation at
, then it is represented by the restriction of
to the boundary of
. More precisely
, where
corresponds to the simplex spanned by the corresponding vectors, and the local orientation in
at
corresponding to the image of the fundamental class in
is given by
. But this is the negative of the local orientation of
given by the standard basis. This finishes the proof and explains why we took the negative orientation in our construction of the induced orientation in terms of an atlas.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
9 Orientation of products
Given two oriented manifolds there is an obvious way to orient their product by choosing the product atlas. If is smooth and we have given orientations as tangential orientations, we note that
is isomorphic to
and the isomorphism is induced by the differential of the projections and then the product orientation is given by the juxtaposition of the orientations of
and
.
Similarly if and
are oriented by a continuos family of local homological orientations, we note that
is isomorphic to
, this isomorphism from the latter to the first is given by the cross product
. By definition of the cross product of the local homological orientation given by the standard basis of
with the local homological orientation given by the standard basis of
is the local homological orientation given by the standard basis of
. Thus the different concepts of product orientations given by the product of an atlas and by the product of local homological orientations agree also.
As a consequence for compact oriented manifolds equipped with fundamental classes the cross product of the fundamental classes corresponds to the product of the orientations induced by the fundamental classes.
10 Orientation of complex manifold
An -dimensional complex manifold is a topological manifold together with an atlas
such that the coordinate changes are holomorphic maps. Given such an atlas the charts considered as maps to
have orientation preserving coordinate changes, since a complex matrix considered as a real matrix has determinant
, the square of the norm of the complex determinant. Thus a complex manifold considered as a real manifold has this way a canonical orientation.
11 References
- [Bredon1993] G. E. Bredon, Topology and geometry, Graduate Texts in Mathematics 139, Springer-Verlag, 1993. MR1224675 (94d:55001) Zbl 0934.55001
- [Dold1995] A. Dold, Lectures on algebraic topology, Springer-Verlag, 1995. MR1335915 (96c:55001) Zbl 0872.55001
- [Greenberg&Harper1981] M. J. Greenberg and J. R. Harper, Algebraic topology, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, 1981. MR643101 (83b:55001) Zbl 0498.55001
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
12 External links
- The Encylopedia of Mathematics article on orientation.
- The Wikipedia page on orientability.
- The Wikipedia page on orientations of vector spaces.
![\epsilon : M \to \{ \pm 1 \}](/images/math/4/6/c/46c2b13b508393e2ed97a260b9023207.png)
2 Orientation of topological manifolds
An orientation of a topological manifold is a choice of a maximal atlas, such that the coordinate changes are orientation preserving. To make this precise we have to define when a homeomorphism from an open subset of
to another open subset
is orientation preserving. We do this in terms of singular homology groups.
Definition 2.1. A homeomorphism from an open subset
of
to another open subset
is orientation preserving, if for each
the map
is the identity map. Here the isomorphisms
is the following: We first take the map to
(or to
) induced by the translation mapping
to
resp.
and then the inverse of the excision isomorphism. The isomorphism
is the inverse of the corresponding map.
Definition 2.2. An orientation of an -dimensional topological manifold
is the choice of a maximal oriented atlas. Here an atlas
is called oriented if all coordinate changes
are orientation preserving.
Note that any oriented atlas defines a maximal oriented atlas by adding all charts such that the atlas is still oriented. This is normally the way an oriented atlas is given.
A topological manifold is called orientable if it has a topological orientation, otherwise it is called non-orientable.
A topological manifold together with a topological orientation is called an oriented topological manifold.
An open subset of an oriented topological manifold is oriented by restring the charts in the maximal oriented atlas to the intersection with the open subset. A homeomorphism between oriented topological manifolds is orientation preserving if for each chart
in the oriented atlas of
the chart
is in the oriented atlas of
.
There are several equivalent formulations of orientations both for topological manifolds and for smooth manifolds which we will explain in the following sections.
3 Reformulation in terms of local homological orientations
An orientation of an -dimensional topological manifold
can also be defined in terms of the local homology groups
for each
in
.
Definition 3.1. A local homological orientation of an -dimensional topological manifold
is the choice of a generator
of the local homology group
for each
. Such a choice is called continuous, if for each
there is an open neighborhood
and a class
such that the map induced by the inclusion
maps
to
for each
. A homological orientation of
is a continuous choice of local homological orientations.
As above an open subset of
has an induced homological orientation which is given by the image under the inverse of the isomorphism induced by the inclusion
.
To get an example consider a finite dimensional oriented real vector space , i.e.
is equipped with an equivalence class of bases
, where two bases are called equivalent, if and only if the matrix of the base change matrix has positive determinant. The orientation of
as a vector space gives a homological orientation of
as a topological space as follows. We first orient at
in
by considering the simplex spanned by
. This contains
in its interior and is a generator of
. By translations we define local orientations at arbitrary points of
mapping the local orientation at
to the local orientation at
by the map induced by the translation mapping
to
. By construction this is a continuous family of local homological orientations and so gives a homological orientation of
. From this we obtain homological orientations of all open subsets of
.
The equivalence of these two concepts of an orientation of a topological manifold is shown as follows. A homeomorphism between manifolds equipped with a continuous family of local orientations is called orientation preserving if the induced map maps the corresponding local orientations to each other. We note that if both manifolds are open subsets of , this definition of orientation preserving homeomorphisms agrees with the one defined above. With this one defines for a topological manifold with a continuous family of local orientations a maximal oriented atlas by all charts which are orientation preserving, where we orient
as above. In turn if one has a maximal oriented atlas one uses it to transport the local orientations of open subsets of
to local homological orientations of
, which are a continuous family, since the atlas is oriented.
4 Orientation of smooth manifolds
The definition of an orientation for a topological manifold needs homology groups. For smooth manifolds the definition can be simplified. To distinguish the very similar definition we call it a smooth orientation.
Definition 4.1. A smooth orientation of an -dimensional smooth manifold
is the choice of a maximal oriented atlas. Here a smooth atlas
is called oriented if the determinant of the derivatives of all coordinate changes
is positive. The oriented atlas is called maximal if it cannot be enlarged to an oriented atlas by adding another chart. Note that any oriented atlas defines a maximal oriented atlas by adding all charts such that the atlas is still oriented. This is normally the way an oriented atlas is given.
A smooth manifold is called orientable if it has a smooth orientation, otherwise it is called non-orientable.
A smooth manifold together with a smooth orientation is called an oriented smooth manifold.
5 From tangential orientations to homological orientations
For smooth manifolds we have now two definitions of an orientation, the smooth orientation and the orientation as a topological manifold. Here we explain why they are again equivalent concepts. The key observation is the following. If we have an orientation of the vector space we have defined corresponding local orientations. If we change the orientation of the vector space
, the local homological orientation changes its sign. Since there are two orientations of
as a vector space and two generators of
this correspondence is a bijection.
The next observation is that if we have a diffeomorphism from an open subset in
to another open subset
in
, then its differential preserves the standard orientation of
if and only if it preserves the corresponding local homological orientations and so the underlying homeomorphism is orientation preserving in the sense defined in the beginning of the last section.
Thus, if is smoothly oriented, i.e. is equipped with a maximal oriented smooth atlas, then - forgetting the smooth structure - we obtain an oriented topological atlas and we define the corresponding topological orientation by passing to the maximal oriented topological atlas containing these charts. In turn, if one has a maximal oriented topological atlas the subset of smooth charts in it defines a smooth orientation.
6 Reformulations of orientation for smooth manifolds
There are several equivalent formulations for orientations of smooth manifolds.
Definition 6.1. A tangential orientation of is a continuous choice of an orientation of the tangent space
in the sense of orientations of vector spaces for every point
. Here continuous means that for every
M there is a chart
around
, such that the differential of
maps for all
the orientation at
to the same orientation of
.
The relation between these two definitions is the following. If is an oriented atlas, we define an orientation of
by choosing an oriented chart
around
and define the tangential orientation as the image of the orientation of
under the differential of
. In turn, if a continuous orientation of
for all
is given, one defines a maximal oriented atlas as the atlas consisting of all charts
such that all differentials are orientation preserving, where we equip
with the induced orientation from
equipped with the standard orientation given by the canonical basis. It is easy to check that these constructions are well defined and give equivalent formulations.
Further equivalent formulations, which need a bit more knowledge of vector bundles are:
- An orientation of a smooth n-dimensional manifold is given by the reduction of the structure group
of the tangent bundle
to
, the subgroup of matrices with determinant
. That this is an equivalence is an easy exercise.
- An orientation of a smooth manifold is given by a trivialization (an isomorphism to the trivial bundle) of the exterior bundle . That this is an equivalence is an easy exercise.
Remark 6.2. Since the different concepts of orientations are all equivalent, one normally speaks of an oriented manifold in all cases. We only used the adjectives to make clear that a priory the definitions are different.
7 Criteria for orientability
There are various criteria for orientability:
Theorem 7.1. A smooth -dimensional manifold is orientable if and only if the tangent bundle (or the normal bundle of an embedding into
) has a Thom class, i.e. a class
, whose restriction to each fibre
is a generator of
. Moreover the choice of a Thom class determines an orientation and vice versa.
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![TM_x](/images/math/b/8/b/b8b036f2add61069375cfe6ac8f65851.png)
![TM](/images/math/8/e/4/8e4f2fdbb13a819d0f3d6796804f9ef4.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 7.2. A smooth manifold is orientable if and only if the first Stiefel Whitney class of its tangent bundle vanishes.
See [Milnor&Stasheff1974, Lemma 11.6 and Problem 12-A] and [Bredon1993, Proposition 17.2].
Theorem 7.3 [Dold1995, VIII Corollary 3.4]. A connected closed -dimensional manifolds
is
orientable if and only if
is non-zero, in which case it
is isomorphic to
. The choice of a generator is called a fundamental class
. The choice of
a generator corresponds to the choice of an orientation [Dold1995, VIII Definition 4.1]. For
a not necessarily connected compact oriented manifold
the components
are oriented and the sum of the fundamental classes of the components
define the fundamental class of
.
There is a generalization of Theorem 7.3 to non-compact manifolds.
Theorem 7.4 [Greenberg&Harper1981, Corollary 22.26]. If is arbitrary, then
is orientable if and only if for each
compact connected subset
there is a class
, such that for each
the map induced by the inclusion
maps
to a generator of
and the classes
mapped to each other under the maps induced by the inclusion
for all
.
The images of the classes
in
define a
homological orientation of
and in turn a homological orientation
determines the classes
.
8 Manifolds with boundary
For manifolds with boundary an orientation is defined as an orientation of its interior. An orientation of
induces an orientation on the boundary
. If
is
-dimensional, we orient the boundary, which is
-dimensional by attaching to
the local orientation
, if the restriction of a chart around
from
to
to the interior is in the oriented atlas of the interior of
. Otherwise we define
. For example it we orient the interval
by the atlas of the interior given by the identity map, then
and and
.
If the dimension of is positive, we define the induced orientation both for smooth or topological manifolds in terms of an induced maximal oriented atlas of the boundary. If
is a (smooth) chart around a boundary point
, such that its restriction to the inner is in the oriented atlas of the inner of
, then the restriction of this chart to
is a chart of
and these charts form a maximal oriented (smooth) atlas of
. The orientation given by this atlas is called the induced orientation on
.
The convention, that we consider the negative orientation on the boundary is for smooth manifolds equivalent to choosing an identification of the restriction of the tangent bundle of to
with
, where we identify
with a subbundle by selecting the outward normal vector field. With other words for smooth manifolds the induced orientation is characterized as the orientation of
, such that any outward pointing normal vector plus this orientation is the given orientation of
.
As for compact manifolds without boundary one can see that a compact connected manifold with boundary is orientable if and only if
is non-zero, in which case it is again isomorphic to
, [Dold1995]. The choice of a generator is called a relative fundamental class and again this fixes an orientation of
.
Our at the first glance slightly ad libitum looking convention is made in such a way that the following holds:
Theorem 8.1. Let be a compact oriented
-dimensional manifold with boundary. If
is the fundamental class compatible with the orientation, then
is the the fundamental class compatible with the induced orientation of the boundary as defined above.
Since the proof of this result is not in standard text books (to my knowledge), we give it here.
Proof. Since the orientation is given locally (we use the homological formulation) it is enough to show that if we consider the local orientation of in a chart near the boundary, the boundary operator maps it to the local orientation of
in the restriction of this chart to the boundary. Here we choose the chart in such a way, that the orientation of
corresponds to the standard orientation of
(if not change your atlas by a reflection in
).
Thus we consider and the local orientation given by the standard basis of
. Since we work with the half space we map the simplex constructed by the standard basis with edges
, so that it is spanned instead by
. We denote this simplex by
. The class represented by this simplex in
for some
in the inner of the simplex is the same as that of
given by the standard orientation of
. If we begin with the fundamental class
, consider its image under the boundary operator in
and pass to the local orientation at
, then it is represented by the restriction of
to the boundary of
. More precisely
, where
corresponds to the simplex spanned by the corresponding vectors, and the local orientation in
at
corresponding to the image of the fundamental class in
is given by
. But this is the negative of the local orientation of
given by the standard basis. This finishes the proof and explains why we took the negative orientation in our construction of the induced orientation in terms of an atlas.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
9 Orientation of products
Given two oriented manifolds there is an obvious way to orient their product by choosing the product atlas. If is smooth and we have given orientations as tangential orientations, we note that
is isomorphic to
and the isomorphism is induced by the differential of the projections and then the product orientation is given by the juxtaposition of the orientations of
and
.
Similarly if and
are oriented by a continuos family of local homological orientations, we note that
is isomorphic to
, this isomorphism from the latter to the first is given by the cross product
. By definition of the cross product of the local homological orientation given by the standard basis of
with the local homological orientation given by the standard basis of
is the local homological orientation given by the standard basis of
. Thus the different concepts of product orientations given by the product of an atlas and by the product of local homological orientations agree also.
As a consequence for compact oriented manifolds equipped with fundamental classes the cross product of the fundamental classes corresponds to the product of the orientations induced by the fundamental classes.
10 Orientation of complex manifold
An -dimensional complex manifold is a topological manifold together with an atlas
such that the coordinate changes are holomorphic maps. Given such an atlas the charts considered as maps to
have orientation preserving coordinate changes, since a complex matrix considered as a real matrix has determinant
, the square of the norm of the complex determinant. Thus a complex manifold considered as a real manifold has this way a canonical orientation.
11 References
- [Bredon1993] G. E. Bredon, Topology and geometry, Graduate Texts in Mathematics 139, Springer-Verlag, 1993. MR1224675 (94d:55001) Zbl 0934.55001
- [Dold1995] A. Dold, Lectures on algebraic topology, Springer-Verlag, 1995. MR1335915 (96c:55001) Zbl 0872.55001
- [Greenberg&Harper1981] M. J. Greenberg and J. R. Harper, Algebraic topology, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, 1981. MR643101 (83b:55001) Zbl 0498.55001
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
12 External links
- The Encylopedia of Mathematics article on orientation.
- The Wikipedia page on orientability.
- The Wikipedia page on orientations of vector spaces.