View source for Orientation

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|Manifold]]; [[Vector bundle|Vector bundle]]; [[Poincaré complex|Poincaré complex]], etc.). The modern view of orientation is given in [[Generalized cohomology theories|Generalized cohomology theories]].

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o0702001.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o0702002.png" /> with complex basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o0702003.png" />, a real basis is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o0702004.png" />, considering the space as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o0702005.png" />. Any two real bases arising in this way from complex bases are positively related (i.e. a complex structure defines an orientation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o0702006.png" />).

In a line, plane or, generally, a real affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o0702007.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o0702008.png" /> connecting the given systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o0702009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020010.png" /> and depending continuously on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020011.png" /> exists. Reflection in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020012.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020013.png" /> 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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020014.png" /> (along with the given rule) defines an orientation. For example, in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020015.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020016.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020017.png" /> is given, then every half-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020018.png" /> defines an orientation on the boundary plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020019.png" />. For example, it may be agreed that if the last <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020020.png" /> vectors in an oriented basis lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020021.png" />, while the first vector points outwards from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020022.png" />, then the last <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020023.png" /> vectors define the orientation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020024.png" />. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020025.png" /> an orientation can be defined by the order of the vertices of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020026.png" />-dimensional simplex (a triangle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020027.png" />, a tetrahedron in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020028.png" />). 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020029.png" />-face <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020030.png" /> of an oriented simplex has an induced orientation: If the first vertex does not belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020031.png" />, then the order of the others is taken to be positive for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020032.png" />.

In a connected manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020033.png" />, the coordinate system takes the form of an atlas: A set of charts (cf. [[Chart|Chart]]) which cover <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020034.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020035.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020036.png" /> (connected charts which contain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020037.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020038.png" /> (for example, a rotation direction on the circle can be defined by choosing one tangent vector). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020039.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020040.png" />, the first vector of which is directed from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020041.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020042.png" />, a chain of charts can be chosen such that two neighbouring charts are positively connected. Thus, an orientation at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020043.png" /> defines an orientation at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020044.png" />, and this relation depends on the path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020045.png" /> only up to its continuous deformation when its ends are fixed. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020046.png" /> is a loop, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020047.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020048.png" /> is called an orientation-reserving loop if these orientations are opposite. A homomorphism of the fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020049.png" /> into a group of order 2 arises: The orientation-reversing loops are sent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020050.png" />, while the others are sent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020051.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020052.png" /> which is trivial if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020053.png" /> is orientable. For a differentiable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020054.png" /> it can be defined as the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020055.png" /> of differential forms of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020056.png" />. It has a non-zero section only in the orientable case and then such a section simultaneously defines a volume form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020057.png" /> and an orientation. This bundle has a classifying mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020058.png" />. The manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020059.png" /> is orientable if and only if the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020060.png" /> which is the image of the class dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020061.png" />, is not equal to zero. It is dual to a cycle whose support is the manifold which is the pre-image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020062.png" /> under the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020063.png" />, taken in [[General position|general position]]. This cycle is said to be orienting, since its complement is orientable: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020064.png" /> is cut by means of the cycle, then an orientable manifold is obtained. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020065.png" /> is itself orientable (non-orientable) if and only if a disconnected manifold (a connected complement) is obtained after the cut. For example, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020066.png" />, a projective line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020067.png" /> serves as orienting cycle.

A triangulated manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020068.png" /> (or a pseudo-manifold) is orientable if it is possible to orient all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020069.png" />-dimensional simplices such that two simplices with a common <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020070.png" />-dimensional face induce opposite orientations on the face. A closed chain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020071.png" />-dimensional simplices each two neighbours of which have a common <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020072.png" />-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|homology group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020073.png" /> (with closed supports) is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020074.png" />, and the choice of one of the two generators defines an orientation. This is also true for a connected manifold with boundary, using <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020075.png" />. In the first instance, orientability is a homotopy invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020076.png" />, while in the second, of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020077.png" />. So, the [[Möbius strip|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020078.png" />, isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020079.png" />. The homological interpretation of orientation enables this concept to be applied to generalized homology manifolds (cf. [[Homology manifold|Homology manifold]]).

Let a fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020080.png" /> with standard fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020081.png" /> be defined uniquely over a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020082.png" />. If the orientation of all fibres can be chosen such that any (non-singular) mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020083.png" />, defined by the path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020084.png" /> 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|completely-continuous operator]].

==Orientation in generalized cohomology theories.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020085.png" /> be a multiplicative generalized cohomology theory (hereafter, simply a theory). There is a unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020086.png" /> for which, given the [[Suspension|suspension]] isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020087.png" />, there is a corresponding element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020088.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020089.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020090.png" />-dimensional sphere.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020091.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020092.png" />-dimensional vector bundle over an arcwise-connected space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020093.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020094.png" /> be the [[Thom space|Thom space]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020095.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020096.png" /> be a standard imbedding, i.e. a homeomorphism on the "fibre" over a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020097.png" />. The element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020098.png" /> is called an orientation or a [[Thom class|Thom class]] of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o07020099.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200100.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200101.png" /> is an invertible element (for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200102.png" />). A bundle possessing an orientation is orientable in the theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200103.png" /> or simply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200105.png" />-orientable, while a bundle with a chosen <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200106.png" />-orientation is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200108.png" />-oriented. The [[Thom isomorphism|Thom isomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200109.png" /> is valid (see [[#References|[6]]]). The set of orientations of a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200110.png" />-oriented bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200111.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200112.png" /> is in one-to-one correspondence with the elements of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200113.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200114.png" /> is the group of invertible elements of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200115.png" />.

The trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200116.png" />-dimensional bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200117.png" /> possesses an orientation in any theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200118.png" />, and if two out of the three bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200119.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200120.png" />-orientable, then the third is also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200121.png" />-orientable (see [[#References|[7]]]). Moreover, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200122.png" />-orientability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200123.png" /> entails the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200125.png" />-orientability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200126.png" />.

The concept of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200127.png" />-orientability is also introduced for any bundle in the sense of Hurewicz <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200128.png" />, a fibre of which is homotopically equivalent to a sphere. The cone of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200129.png" /> is called the Thom space of this bundle; further definitions are analogous. The definition of orientation of a vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200130.png" /> reduces to this if a bundle of unit spheres (in some Riemannian metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200131.png" />) associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200132.png" /> is taken as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200133.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200134.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200135.png" />, the property of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200136.png" />-orientability follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200137.png" />-orientability.

==Examples.==


1) In the theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200138.png" />, any vector (sphere) bundle is orientable.

2) In the theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200139.png" />, those bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200140.png" /> for which the Stiefel–Whitney characteristic class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200141.png" /> are orientable, i.e. those bundles which are orientable in the classical sense.

3) The orientability of a vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200142.png" /> in real [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200143.png" />-theory]] is equivalent to the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200144.png" />, while in complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200145.png" />-theory it is equivalent to the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200146.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200147.png" /> is an integral class [[#References|[8]]]. For sphere bundles to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200148.png" />-orientable, this condition is necessary, though not sufficient.

4) In the theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200149.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200150.png" /> act on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200151.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200152.png" /> be a certain theory. A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200153.png" /> with a universal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200154.png" />-oriented bundle over it exists (see [[#References|[7]]], where an explicit construction is given) which classifies the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200155.png" />-oriented vector bundles with structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200156.png" />, i.e. for any (arcwise connected) space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200157.png" />, the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200158.png" />-oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200159.png" />-vector bundles over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200160.png" /> is in natural one-to-one correspondence with a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200161.png" /> of homotopy classes of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200162.png" />. This is also true for sphere bundles and "good" monoids <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200163.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200164.png" /> all vector bundles are orientable, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200165.png" /></td> </tr></table>

Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200166.png" />. In this context, the conditions on the theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200167.png" /> are weakened, for example, the condition of commutativity of multiplication is dropped, etc. For any theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200168.png" /> in which all complex bundles are orientable, there is a homomorphism of theories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200169.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200170.png" /> is the theory of unitary [[Cobordism|cobordism]], and this homomorphism is completely defined by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200171.png" />-orientation of the canonical bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200172.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200173.png" />. The same is true for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200174.png" />-bundles (see [[Cobordism|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200175.png" /> such that the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200176.png" /> given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200177.png" /> (see [[#References|[9]]]) is an isomorphism, is called an orientation (or fundamental class) of the closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200178.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200179.png" /> (or, more generally, of the Poincaré complex of formal dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200181.png" />) in the theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200182.png" />. This isomorphism is the so-called Poincaré duality isomorphism. A manifold (Poincaré complex) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200183.png" />-orientable if and only if its normal bundle is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200184.png" />-orientable. An orientation is also defined for manifolds (Poincaré complexes) with boundary.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, "Modern geometry" , '''1–2''' , Springer (1984–1985) (Translated from Russian) {{MR|1138462}} {{MR|1076994}} {{MR|1011459}} {{MR|1011458}} {{MR|0864355}} {{MR|0822730}} {{MR|0822729}} {{MR|0807945}} {{MR|0766739}} {{MR|0736837}} {{MR|0566582}} {{ZBL|0751.53001}} {{ZBL|0703.55001}} {{ZBL|0601.53001}} {{ZBL|0565.57001}} {{ZBL|0582.55001}} {{ZBL|0529.53002}} {{ZBL|0433.53001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> , ''Introduction to topology'' , Moscow (1980) (In Russian) {{MR|}} {{ZBL|0478.57001}} {{ZBL|1081.54501}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V.A. Rokhlin, D.B. Fuks, "Beginner's course in topology. Geometric chapters" , Springer (1984) (Translated from Russian) {{MR|759162}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) {{MR|0229247}} {{ZBL|0144.44804}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) {{MR|0210112}} {{MR|1325242}} {{ZBL|0145.43303}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> A. Dold, "Relations between ordinary and extraordinary homology" , ''Colloq. Algebraic Topology, August 1–10, 1962'' , Inst. Math. Aarhus Univ. (1962) pp. 2–9 {{MR|}} {{ZBL|0145.20104}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> J.P. May, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200185.png" />-ring spaces and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200186.png" />-ring spectra" , ''Lect. notes in math.'' , '''577''' , Springer (1977) {{MR|0494077}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) {{MR|0248858}} {{ZBL|0181.26604}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> G.W. Whitehead, "Recent advances in homotopy theory" , Amer. Math. Soc. (1970) {{MR|0309097}} {{ZBL|0217.48601}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> Yu.B. Rudyak, "On the orientability of spherical, topological, and piecewise-linear fibrations in complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070200/o070200187.png" />-theory" ''Soviet Math. Dokl.'' , '''37''' : 1 (1988) pp. 283–286 ''Dokl. Akad. Nauk SSSR'' , '''298''' : 6 (1988) pp. 1338–1340</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M.W. Hirsch, "Differential topology" , Springer (1976) {{MR|0448362}} {{ZBL|0356.57001}} </TD></TR></table>