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Orientability
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A surface S in the Euclidean space R3 is orientable if a two-dimensional figure (for example, ) cannot be moved around the surface and back to where it started so that it looks like its own mirror image ().
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A surface S in the Euclidean space R3 is orientable if a two-dimensional figure (for example, ) cannot be moved around the surface and back to where it started so that it looks like its own mirror image (). Otherwise the surface is non-orientable.
More precisely, and applicable to non-embedded surfaces, a surface is orientable if there is no continuous map f from the product of a 2-dimensional ball B and the unit interval [0,1] to the surface, such that f(b,t) = f(c,t) only if b = c for every t in [0,1], and there exists a reflection map r such that f(b,0) = f(r(b),1) for every b in B.
An abstract surface (i.e., a two-dimensional manifold) is orientable if a consistent concept of clockwise rotation can be defined on the surface in a continuous manner. This turns out to be equivalent to the question of whether the surface contains no subset that is homeomorphic to the Möbius strip. Thus, for surfaces, the Möbius strip may be considered the source of all non-orientability.
A surface that is embedded in R3 will be orientable in the sense if and only if it is orientable as an abstract surface.
Oriented versus orientable
For an orientable surface, a consistent choice of "clockwise" (as opposed to counter-clockwise) is called an orientation, and the surface is called oriented. An orientable surface admits exactly 2 orientations, and the distinction between an oriented surface and an orientable surface is subtle and frequently blurred. An orientable surface is an abstract surface that admits an orientation, while an oriented surface is a surface that is abstractly orientable, and has the additional datum of a choice of one of the 2 possible orientations.
Examples
Most surfaces we encounter in the physical world are orientable. Spheres, planes, and tori are orientable, for example. But Möbius strips, real projective planes, and Klein bottles are non-orientable. They, as visualized in 3-dimensions, all have just one side. (Caveat: the real projective plane and Klein bottle can't be embedded in R3, only immersed with nice intersections.)
Note that locally an embedded surface always has two sides, so a near-sighted ant crawling on a one-sided surface would think there is an "other side". The essence of one-sidedness is that the ant can crawl from one side of the surface to the "other" without going through the surface or flipping over an edge, but simply by crawling far enough.
In general, the property of being orientable is not equivalent to being two-sided; however, this holds when the ambient space (such as R3 above) is orientable. For example, a torus embedded in can be one-sided, and a Klein bottle in the same space can be two-sided; here refers to the Klein bottle.
A simply connected two-dimensional space which obeys Euclidean geometry is orientable.
The space-time manifold of the actual universe is believed to be orientable.Otherwise, you could take a round trip along some noncontractible path through spacetime, then when you arrived back you (or the rest of the universe, from your perspective) would have become left-right reversed, like a mirror image of itself (see chirality and handedness).
Orientation by a triangulation
Orientability, for surfaces, is easily defined, regardless of whether the surface is embedded in an ambient space or not. Any surface has a triangulation: a decomposition into triangles such that each edge on a triangle is glued to at most one other edge. We can orient each triangle, by choosing a direction for each edge (think of this as drawing an arrow on each edge) so that the arrows go from head to tail as we go around the boundary of the triangle. If we can do this so that in addition triangles sharing an edge have arrows on that edge going in the same direction, then what we've done is chosen an orientation for the surface. In fact, such a choice will only be possible if the surface is orientable and moreover, if it is orientable then there will exist exactly two different orientations. Note that whether the surface is orientable is independent of triangulation; this fact is not obvious, but a standard exercise.
This rather precise definition is based on intuition gathered from observing the following phenomenon:
Imagine a figure on the surface, that can freely slide along the surface but cannot be lifted off the surface (figure is chosen because of its handedness). If the surface is a Möbius band, and the figure slides all the way around the band and returns to its starting point, then it will look like mirror-image rather than . If the surface is a sphere, on the other hand, that cannot happen.
The relation to the definition above is that sliding the around from triangle to triangle in a triangulation gives an orientation for each triangle; the in a triangle induces a choice of arrow for each edge, based on the order red-green-blue of colors. The only obstruction to consistently orienting all the triangles is that when the returns to its original starting triangle, it may induce choices of arrows going opposite to the original choice. Clearly, if this never happens, then we want the surface to be orientable, whereas if this does happen, then we want to call the surface non-orientable.
The definition above can be generalized to an n-manifold that has a triangulation, but there are problems with that approach: some 4-manifolds do not have a triangulation, and in general for n > 4 some n-manifolds have triangulations that are inequivalent.
Orientability of manifolds
Topological definitions
An n-dimensional manifold (either embedded in a finite dimensional vector space, or an abstract manifold) is called non-orientable if it is possible to take the homeomorphic image of an n-dimensional ball in the manifold and move it through the manifold and back to itself, so that at the end of the path, the ball has been reflected, using the same definition as for surfaces above. Equivalently, a n-dimensional manifold is non-orientable if it contains a homeomorphic image of the space formed by taking the direct product of a (n-1)-dimensional ball B and the unit interval [0,1] and gluing the ball B× at one end to the ball B× at other end with a single reflection. For surfaces, this space is a Möbius strip; for 3-manifolds, this is a solid Klein bottle.
As another alternative definition, in the language of structure groups, an orientable manifold is one whose structure group (a priori GL(n)) can be reduced to the subgroup GL+(n) of orientation-preserving transforms. Concretely, an orientable manifold is one that has a cover of open n-dimensional balls with consistent orientations (i.e. all transition maps are orientation preserving). Here one needs to define what a local orientation means, which can be done using orientations of vector bundles (a local orientation is an orientation of the tangent spaces at a point) or using singular homology (an orientation is a choice of generator of the n-th relative homology group
at a point p). A manifold is then said to be orientable if one can choose local orientations consistently throughout the manifold.
Using homology allows one to define orientability for compact n-manifolds without considering local orientations. A compact n-manifold M is orientable if and only if the top homology group, , is isomorphic to . Considering simplicial homology, which applies to any triangulable manifold, allows one to consider this a concrete statement about coherently orienting top-dimensional simplices in a triangulation, as done in the surface case above.
If the manifold has a differentiable structure, one can use the language of differential forms (see below).
Orientation of differential manifolds by top-dimensional forms
Another way of thinking about orientability is thinking of it as a choice of "right handedness" vs. "left handedness" at each point in the manifold.
Formally, a -dimensional differentiable manifold is called orientable if it possesses a differential form of degree (i.e volume form) which is nonzero at every point on the manifold. Conversely, given such a form , we say that the manifold is oriented by .
The crucial point to observe here is that such a differential form gives a choice of "right-handed" basis at each point. A traveler in an orientable manifold will never change his/her handedness by going on a round trip.
Orientable double cover
A closely related notion uses the idea of covering space. For a connected manifold M take M*, the set of pairs (x, o) where x is a point of M and o is an orientation at x; here we assume M is either smooth so we can choose an orientation on the tangent space at a point or we use singular homology to define orientation. Then for every open, oriented subset of M we consider the corresponding set of pairs and define that to be an open set of M*. This gives M* a topology and the projection sending (x, o) to x is then a 2-1 covering map. This covering space is called the orientable double cover, as it is orientable. M* is connected if and only if M is not orientable.
Another way to construct this cover is to divide the loops based at a basepoint into either orientation-preserving or orientation-reversing loops. The orientation preserving loops generate a subgroup of the fundamental group which is either the whole group or index two. In the latter case (which means there is an orientation-reversing path), the subgroup corresponds to a connected double covering; this cover is orientable by construction. In the former case, one can simply take two copies of M, each of which corresponds to a different orientation.
Orientation of vector bundles
A real vector bundle, which a priori has a GL(n) structure group, is called orientable when the structure group may be reduced to , the group of matrices with positive determinant. This reduction is always possible if the underlying base manifold is orientable and in fact this provides a convenient way to define the orientability of a smooth real manifold: a smooth manifold is defined to be orientable if its tangent bundle is orientable (as a vector bundle). Note that as a manifold in its own right, the tangent bundle is always orientable, even over nonorientable manifolds.
Related concepts
The notion of orientability is essentially derived from the topology of the real general linear group , specifically that the lowest homotopy group is : an invertible transform of a real vector space is either orientation-preserving or orientation-reversing.
This holds not only for differentiable manifolds but for topological manifolds, as the space of self-homotopy equivalences of a sphere also has two connected components, which can be denoted the "orientation-preserving" and "orientation-reversing" maps.
The analogous notion for the symmetric group is the alternating group of even permutations.
See also
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