Orientation

2020 Mathematics Subject Classification: Primary: 54-XX [MSN][ZBL]

The notion of Orientation is a formalization and far-reaching generalization of the concept of direction on a curve. The orientation of special classes of spaces is defined (cf. Manifold; Vector bundle; Poincaré complex, etc.). The modern view of orientation is given in Generalized cohomology theories.

General

In classical mathematics, an orientation is the choice of an equivalence class of coordinate systems, where two coordinate systems belong to the same class if they are positively related (in a specific sense).

In the case of a finite-dimensional vector space $\R^n$, a coordinate system is given by a basis, and two bases are positively related if the determinant of the transition matrix from one to the other is positive. There are two classes here. In a complex space $\C^n$ with complex basis $e_1,\dots,e_n$, a real basis is given by $e_1,\dots,e_n,ie_1,\dots,ie_n$, considering the space as $\R^{2n}$. Any two real bases arising in this way from complex bases are positively related (i.e. a complex structure defines an orientation on $\R^{2n}$).

In a line, plane or, generally, a real affine space $E^n$, a coordinate system is given by the choice of a point (origin) and a basis. The change of coordinates is defined by a translation (changing the origin) and a change of basis. This change is positive if the matrix of the base change has positive determinant. (For example, an even permutation of the vectors in the basis.) Two coordinate systems define the same orientation if one of them can be continuously transformed into the other, i.e. if a family of coordinate systems $O_t, e_t$ connecting the given systems $O_0, e_0$ and $O_1, e_1$ and depending continuously on $t\in[0,1]$ exists. Reflection in an $(n-1)$-dimensional plane gives the opposite orientation, i.e. the other class.

Classes of coordinate systems can be defined by different geometric figures. If a figure $X$ is related by a specific rule to a coordinate system, then its mirror image should be related by the same rule to a coordinate system with the opposite orientation. In this way, $X$ (along with the given rule) defines an orientation. For example, in the plane $E^2$, a circle with a given direction of traversal defines a coordinate system from one class by the rule that the origin is at the centre of the circle, with the first basis vector taken arbitrarily while the second is taken so that the rotation from the first to the second through the smaller angle is the direction of traversal on the circle. In $E^3$, a frame can be related to a screw. The first vector goes in the direction the screw moves when being screwed in, and the rotation from the second vector to the third coincides with the rotation of the screw as it is screwed in (it is supposed that all screws are threaded in the same way). A basis (frame) can also be defined in a well-known way by using the thumb and first two fingers on one's hand, as in the right-hand rule.

If an orientation of $E^n$ is given, then every half-space $E_+^n$ defines an orientation on the boundary plane $E^{n-1}$. For example, it may be agreed that if the last $n-1$ vectors in an oriented basis lie in $E^{n-1}$, while the first vector points outwards from $E_+^n$, then the last $n-1$ vectors define the orientation of $E^{n-1}$. In $E^n$ an orientation can be defined by the order of the vertices of an $n$-dimensional simplex (a triangle in $E^2$, a tetrahedron in $E^3$). A basis is defined by choosing the origin at the first vertex, while the vectors of the basis point to the other vertices. Two orders define the same orientation if and only if they differ by an even permutation. A simplex with a fixed order of vertices up to an even permutation is said to be oriented. Every $(n-1)$-face $\def\s{\sigma}\s^{n-1}$ of an oriented simplex has an induced orientation: If the first vertex does not belong to $\s^{n-1}$, then the order of the others is taken to be positive for $\s^{n-1}$.

In a connected manifold $M$, the coordinate system takes the form of an atlas: A set of charts (cf. Chart) which cover $M$. The atlas is said to be orienting if the coordinate transformations between charts are all positive. In the case of a differentiable manifold this means that the Jacobians of the coordinate transformations between any two charts are positive at every point. If an orienting atlas exists, then $M$ is orientable. In this case, all orienting atlases divide into two classes such that the transition from the charts of one atlas to the charts of another is positive if and only if both atlases belong to the same class. A choice of this class is called an orientation of the manifold. This choice can be made by choosing one chart or local orientation at a point $x_0$ (connected charts which contain $x_0$ naturally divide into two classes). In the case of a differentiable manifold, a local orientation can be defined by choosing a basis in the tangent plane at the point $x_0$ (for example, a rotation direction on the circle can be defined by choosing one tangent vector). If $M$ has a boundary and is oriented, then the boundary is also orientable, for example according to the rule: At a point of the boundary, a basis is taken which orients $M$, the first vector of which is directed from $\partial M$, while the other vectors lie in the tangent plane to the boundary; these latter vectors are taken to be an orienting basis of the boundary.

Along any path $q:[0,1]\to M$, a chain of charts can be chosen such that two neighbouring charts are positively connected. Thus, an orientation at the point $q(0)$ defines an orientation at the point $q(1)$, and this relation depends on the path $q$ only up to its continuous deformation when its ends are fixed. If $q$ is a loop, i.e. $q(0)=q(1)=x_0$, then $q$ is called an orientation-reserving loop if these orientations are opposite. A homomorphism of the fundamental group $\pi_1(M,x_0)$ into a group of order 2 arises: The orientation-reversing loops are sent to $-1$, while the others are sent to $+1$. Through this homomorphism a covering is created, which is a two-sheeted covering in the case of a non-orientable manifold. It is said to be orienting (since the covering space will be orientable). This same homomorphism defines a line bundle over $M$ which is trivial if and only if $M$ is orientable. For a differentiable $M$ it can be defined as the bundle $\def\L{\Lambda}\L^n(M)$ of differential forms of order $n$. It has a non-zero section only in the orientable case and then such a section simultaneously defines a volume form on $M$ and an orientation. This bundle has a classifying mapping $k:M\to \R P^n$. The manifold $M$ is orientable if and only if the class $\mu\in H^{n-1}(M;\Z)$ which is the image of the class dual to $\R P^{n-1} \subset \R P^n$, is not equal to zero. It is dual to a cycle whose support is the manifold which is the pre-image of $\R P^{n-1}$ under the mapping $k$, taken in general position. This cycle is said to be orienting, since its complement is orientable: If $M$ is cut by means of the cycle, then an orientable manifold is obtained. $M$ is itself orientable (non-orientable) if and only if a disconnected manifold (a connected complement) is obtained after the cut. For example, in $\R P^{2}$, a projective line $\R P^1$ serves as orienting cycle.

A triangulated manifold $M$ (or a pseudo-manifold) is orientable if it is possible to orient all $n$-dimensional simplices such that two simplices with a common $(n-1)$-dimensional face induce opposite orientations on the face. A closed chain of $n$-dimensional simplices each two neighbours of which have a common $(n-1)$-face is said to be orientation-reversing if these simplices can be oriented such that the first and last simplices induce coinciding orientations on the common face, while the other neighbours induce opposite orientations.

An orientation can be defined in the language of homology theory thus: For a connected orientable manifold without boundary, the homology group $H_n(M;\Z)$ (with closed supports) is isomorphic to $\Z$, and the choice of one of the two generators defines an orientation. This is also true for a connected manifold with boundary, using $H_n(M,\partial M;\Z)$. In the first instance, orientability is a homotopy invariant of $M$, while in the second, of the pair $(M,\partial M)$. So, the Möbius strip and the annulus have one and the same homotopy type but a different one if one considers the boundary. A local orientation of the manifold can also be defined by the choice of generators in the group $H_n(M,M\setminus x_0;\Z)$, isomorphic to $\Z$. The homological interpretation of orientation enables this concept to be applied to generalized homology manifolds (cf. Homology manifold).

Let a fibration $p:E\to X$ with standard fibre $F^n$ be defined uniquely over a space $X$. If the orientation of all fibres can be chosen such that any (non-singular) mapping $\def\g{\gamma} p^{-1}(\g(0))\to p^{-1}(\g(1))$, defined by the path $\g:(0,1)\to X$ up to a non-singular homotopy, preserves the orientation, then the fibration is oriented, while the choice of the orientation of the fibres is the orientation of the fibration. For example, a Möbius strip, looked at as a vector bundle over a circle, does not possess an orientation, whereas the lateral surface of a cylinder does.

The concept of orientation also allows a natural generalization for the case of an infinite-dimensional manifold modelled on an infinite-dimensional Banach or topological vector space. This requires restrictions on the linear operators which are differentials of transitions from one chart to another: They must not simply belong to the general linear group of all isomorphisms of the structure space, which is homotopically trivial (in the uniform topology) for the majority of classical vector spaces, but must also be contained in a disconnected subgroup of the general linear group. The connected component of the given subgroup will then also provide the "sign" of the orientation. The subgroup usually chosen is the Fredholm group, consisting of those isomorphisms of the structure space for which the difference from the identity isomorphism is a completely-continuous operator.

Orientation in generalized cohomology theories.

Let $E^*$ be a multiplicative generalized cohomology theory (hereafter, simply a theory). There is a unit $\def\t#1{\widetilde{#1}}i\in\t{E}^n(S^n) $ for which, given the suspension isomorphism $\t{E}^0(S^0)\approx \t{E}^n(S^n)$, there is a corresponding element $\g\in \t{E}^n(S^n)$, where $S^n$ is the $n$-dimensional sphere.

Let $\xi$ be an $n$-dimensional vector bundle over an arcwise connected space $X$ and let $T\xi$ be the Thom space of $\xi$. Let $i:S^n\to T\xi$ be a standard imbedding, i.e. a homeomorphism on the "fibre" over a point $x_0\in X$. The element $u\in \t{E}^n(T\xi)$ is called an orientation or a Thom class of the bundle $\xi$ if $i^*(u) = \def\e{\epsilon}\e \g_n$, where $\e\in\t{E}^0(S^0)$ is an invertible element (for example, $\e = 1$). A bundle possessing an orientation is orientable in the theory $E^*$ or simply $E$-orientable, while a bundle with a chosen $E$-orientation is $E$-oriented. The Thom isomorphism $\t{E}^*(T\xi)\approx E^*(X)$ is valid (see [Do]). The set of orientations of a given $E$-oriented bundle $\xi$ over $X$ is in one-to-one correspondence with the elements of the group $\t{E}^0(X)\oplus (\t{E}^0(S^0))^*$, where $(\cdot)^*$ is the group of invertible elements of the ring $(\cdot)$.

The trivial $n$-dimensional bundle $\theta^n$ possesses an orientation in any theory $E^n$, and if two out of the three bundles $\xi,\eta,\xi\oplus\eta$ are $E$-orientable, then the third is also $E$-orientable (see [Ma]). Moreover, the $E$-orientability of $\xi$ entails the $E$-orientability of $\xi\oplus\theta^n$.

The concept of $E$-orientability is also introduced for any bundle in the sense of Hurewicz $p:M\to B$, a fibre of which is homotopically equivalent to a sphere. The cone of the mapping $p$ is called the Thom space of this bundle; further definitions are analogous. The definition of orientation of a vector bundle $\xi$ reduces to this if a bundle of unit spheres (in some Riemannian metric on $\xi$) associated with $\xi$ is taken as $M$. $E$-orientability is an invariant of the stable fibre-wise homotopy type of a vector (sphere) bundle. A bundle which is orientable in one theory is not necessarily orientable in another, but given a ring homomorphism of theories $E^*\to F^*$, the property of $E$-orientability follows from $F$-orientability.

Examples.

1) In the theory $H^*(-;\Z_2)$, any vector (sphere) bundle is orientable.

2) In the theory $H^*(-;\Z)$, those bundles $\xi$ for which the Stiefel–Whitney characteristic class $w_1(\xi) = 0$ are orientable, i.e. those bundles which are orientable in the classical sense.

3) The orientability of a vector bundle $\xi$ in real $K$-theory is equivalent to the fact that $w_1(\xi) = w_2(\xi) = 0$, while in complex $K$-theory it is equivalent to the fact that $w_1(\xi) = 0$ and $w_2(\xi)$ is an integral class [St]. For sphere bundles to be $K$-orientable, this condition is necessary, though not sufficient.

4) In the theory $\pi^*$ of stable cohomotopy groups, only bundles of trivial stable fibre-wise homotopy type are orientable.

In the problem of describing the class of bundles which are orientable in a given theory, the following general result holds. Let a topological group $G$ act on $\R^n$ and let $E^*$ be a certain theory. A space $B(G,E)$ with a universal $E$-oriented bundle over it exists (see [Ma], where an explicit construction is given) which classifies the $E$-oriented vector bundles with structure group $G$, i.e. for any (arcwise connected) space $X$, the set of $E$-oriented $G$-vector bundles over $X$ is in natural one-to-one correspondence with a set $[X,B(G,E)]$ of homotopy classes of mappings $X\to B(G,E)$. This is also true for sphere bundles and "good" monoids $G$.

The opposite problem consists of describing a theory in which a given bundle (or class of bundles) is orientable. It is known that if in a theory $E^*$ all vector bundles are orientable, then

$$E^*(X)\approx H^*(X; \t{E}(S^0)).$$ Moreover, $2E^*(S^0) = 0$. In this context, the conditions on the theory $E^*$ are weakened, for example, the condition of commutativity of multiplication is dropped, etc. For any theory $E^*$ in which all complex bundles are orientable, there is a homomorphism of theories $U^*\to E^*$, where $U^*$ is the theory of unitary cobordism, and this homomorphism is completely defined by the $E$-orientation of the canonical bundles $\eta$ over $\C P^\infty$. The same is true for ${\bf Sp}$-bundles (see Cobordism). Constructing for a given class of vector bundles the universal theory, which maps onto any other theory in which the class of bundles is orientable, has yet to be carried out (1989).

An element $z\in E_n(M)$ such that the homomorphism $E^i(M) \to E_{n-i}(M)$ given by $x\to z\cap x$ (see [Wh]) is an isomorphism, is called an orientation (or fundamental class) of the closed $n$-dimensional manifold $M$ (or, more generally, of the Poincaré complex of formal dimension $n$) in the theory $E^*$. This isomorphism is the so-called Poincaré duality isomorphism. A manifold (Poincaré complex) is $E$-orientable if and only if its normal bundle is $E$-orientable. An orientation is also defined for manifolds (Poincaré complexes) with boundary.

References

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