orientation

The concept of orientation looks so magical if we think of various formulations all describe exactly the same mathematical phenomenon, while they are all rooted on the classic idea of choice of direction, right or left, plus or minus, or man or woman.

To put out the algebraic view of orientation, consider finite discrete set (can be seen as a set of points in general position in dimensional affine space) and its permutation group . The kernel of induces the unique choice of direction for each permutation. In case of , it coincides with the common sense of orientability: corresponds to the right and corresponds to the left.

For its connection with topology, we think it is important to describe the case when a space is manifold so one can study further variants (e.g. pseudo-manifold, vector bundle, etc.) derived from the case of manifold. It is notable that the orientability is preserved by homeomorphism but not a homotropy invariant (compare mobius band and a circle for example).

Thanks to its locally Euclidian property (and its consistent dimension), the orientation can always be defined locally on topological manifold ; but not for globally over .

Let be a commutative ring with unit. For a topological n-manifold , define a covering space as

Here we restrict to a generator of to see an element of via canonical isomorphism . Then we have a fiber . Topologizing with an open basis

where is a collection of an element such that and is the image of by the canonical map . Then is a covering space with local triviality condition . We always have two sheeted orientable sub-covering space where denotes a generator of (the fiber over is ).

A section of a covering space is a continuous right inverse of covering projection. The orientation is the section with a generator of for each . Note that by construction of the covering space, a section encodes local consistency (of a local orientation) if and only if the section represents the orientation:

A function has local consistency if for each , there exists an open neighborhood of such that for each , is the image of by the map .

Lemma 3.27. Let be a manifold of dimension n and let be a compact subset. Then:

(a) If is a section of the covering space , then there is a unique class whose image in is for all .

(b) for .

A. Hatcher, Algebraic topology, Cambridge Univ. Press, 2000

Let be the set of sections of . Sum and scalar multiplication of sections are again sections, so is a R-module.
The lemma indicates that the homomorphism defined by is an isomorphism.

Suppose is connected, then the evaluation map is injective, since the value of a single point determines the given section uniquely.
For it to be surjective, consider the case when , which is to say when is a closed manifold. The composition

is surjective if a section exists such that each point is sent to a generator where is an unit of (Note that a section always exists for each by setting where is the canonical map).

To reduce the orientability to the number of components in the orientable covering space (deduced from a connected manifold), we think of a lemma regarding the n-folded covering space over a connected manifold:

Lemma. Let be a connected manifold. For n-sheeted covering projection , has n components iff is homeomorphism for any component .

The proof is straightforward.
If is homeomorphism, then for any component , there are no distinct two points such that so each fiber can intersect only once with each component. This shows that the number of components of is n or greater than n. If it is greater than n, there is a component and a point such that . Because is connected, a choice of point in fiber determines the component uniquely (owing to path lifting property), we conclude that , contradiction.
For the reverse direction, if has n components, each point in fiber distributes evenly to each component, implying that is injection (as in previous discussion). Because the fiber cannot be empty, must be surjection onto .

Now that we introduce a remarkable result for a case of closed manifold:

Theorem 3.26. Let be a closed connected n manifold. Then:
(a) If is -orientable, the map is an isomorphism for all .
(b) If is not -orientable, the map is injective with image for all .
(c) for .

A. Hatcher, Algebraic topology, Cambridge Univ. Press, 2000

For the last part of (b), identify with . Since the section of two sheeted covering with coefficient gives homeomorphism, we have for all . Because the image of the map is exactly the sum of such , namely , it follows the result.

It is remarkable that every manifold is -orientable, without assuming that is compact or connected (to remove the without boundary condition, we need to treat collar neighborhood properly, which we would not do in this article). This is because the homeomorphic covering projection can be defined, or the section suffices local consistency since the isomorphism is unique).