Orientation (vector space)

The orientation is a concept from linear algebra and differential geometry. In one -dimensional space two bases have the same orientation if they emerge by linear mappings with positive determinant of the transformation matrix (for example, dilations and rotations ) apart. Reflections are also required, the determinant is negative and the bases are the same orientation.

Clearly, there are two possible orientations, an exchange between the orientations is not possible by rotations. Examples:

• In the plane: Mirror writing has a different orientation than Scripture.
• Watches rotate clockwise clockwise and not counter-clockwise.

• My reflection has a different orientation than I do.
• Screws with right-hand thread have a different orientation than screws with left-hand thread.

It should be noted that the examples of the plane in space do not have a different orientation, because they do not have three-dimensional depth.

• 2.1 Definition ( by means of the tangent space )
• 2.2 Coordinate -free definition
• 2.3 Homological orientation of a manifold
• 2.4 Generalized homology theories

• 3.1 cohomological orientation ( Generalized cohomology theories )

Orientation of a vector space

Introduction

Is a finite dimensional vector space. Then you can run any linear mapping ( such a mapping is called an endomorphism ) represented as ( coordinate ) matrix. This matrix representation is dependent on the choice of basis. Be now and bases of. To transform now from the base into the base, you can always find a change of basis matrix. So it varies under different bases, the representation of the function.

Now we examine the determinant of. This can never be zero because Basiswechselmatrizen are always bijective and thus regular. Takes a positive value, it is said, the bases and have the same orientation. As one can easily consider this definition can not be applied to vector spaces over arbitrary fields, but only those parent bodies.

Definition

The orientation is defined by an equivalence relation between the bases of a vector space. Defining the equivalence relation on the basis of the transformation matrix between two bases, as follows:

With respect to this equivalence relation, there are two equivalence classes. The fact that this equivalence relation is well defined and there are actually only two equivalence classes, ensures the determinants multiplication theorem and the fact that basic transformations are reversible.

Now is called each of these two equivalence classes of orientation. An orientation of a vector space is thus specified by specifying an equivalence class of bases, for example, by specifying an equivalence class associated with this basis.

Each belonging to the selected equivalence class base is then called positively oriented, the others are called negatively oriented.

Example

In both, as well as bases. The basis transformation matrix. The determinant of is. So the two bases are not the same orientation and representatives of the two different equivalence classes.

This can be easily illustrate: First base corresponds to a " normal" coordinate system in which the axis - axis to the right and the "points" to the top. Returning to exactly one of these two axes, that "points" to the left, the axis or the axis downwards, but not both, then obtaining a second base with a different orientation.

Homological and cohomological orientation

For a real - dimensional vector space and the choice of an orientation for is similar to selecting one of the two producers.

But after you look at an embedding of the -dimensional standard simplex, which reflects the barycenter by (and therefore the side surfaces after ). Such a map is a relative Cycles and represents a generator of. Two such embeddings are exactly the same then generators when either or both are both orientation- preserving non- orientation- preserving.

Because too is dual, is defined by an orientation and the associated choice of a producer of a producer of.

Orientation of a manifold

Definition ( by means of the tangent space )

An orientation of a -dimensional differentiable manifold is a family of orientations for each tangent space, which depends continuously on the base point in the following sense:

At each point there is a defined on an open area of map with coordinate functions, ..., so that at each point induced by the map in the tangent basis

Is positive with respect to orientation.

A manifold is orientable, if such an orientation exists. An equivalent characterization of orientability provides the following sentence:

Is orientable if an atlas of exists, such that for all cards with non- empty intersection, and for all in the domain of the following applies:

Herein, the Jacobian matrix.

Coordinate -free definition

Be a smooth -dimensional manifold. This manifold is orientable if there exists a smooth, non- degenerate form.

Homological orientation of a manifold

Be one - dimensional ( topological ) manifold and a ring. With the help of Ausschneidungsaxioms for a homology theory we obtain:

An orientation on a selection of producers

With the following compatibility condition: For each there is an open environment and an element such that for all induced by the inclusion of space pairs figure on the homology

The item on maps. For example, does the concept of orientation in line with the usual orientation term. For other rings, however, one can obtain different results; For example, the construction of each manifold - orientable.

Generalized homology theories

Be a given by a ring spectrum (reduced ) generalized homology theory. We denote the image of under the iterated Einhängungs - isomorphism. For a closed manifold, a point and an open environment is a continuous map which is a homeomorphism on and constant on the complement of. Then is called a homology class

An orientation or fundamental class if

Applies to all. For singular homology, this definition agrees with the above.

Orientation of a vector bundle

An orientation of a vector bundle is a family of orientations for each individual fiber, which depends continuously on the base point in the following sense:

For each point there exists an open neighborhood with a local trivialization, so that for each by

Defined mapping of orientation- preserving is after.

A manifold is orientable so if their tangent bundle is orientable.

Cohomological formulation: applies for an orientable -dimensional vector bundles with zero average and there is a producer of whose restriction corresponds to each of the chosen orientation of the fiber.

The one selected orientation corresponding cohomology class

Ie Thom class or orientation class of the oriented vector bundle.

Alternatively, you can also use the Thom - space whose cohomology is isomorphic to. The Thom class then corresponds to the image of (with respect to cup product ) neutral element under the Thom isomorphism.

Cohomological orientation ( Generalized cohomology theories )

Be a given by a ring spectrum (reduced ) generalized cohomology theory with neutral element. We denote the image of under the iterated Einhängungs - isomorphism. For each of the inclusion induces a map. A cohomological orientation with respect to the cohomology theory is - by definition - an element

With for all.

Examples:

• In the case of singular cohomology with coefficients corresponding to the above definition and is the Thom - class.
• Each vector bundle is orientable with respect to singular cohomology with coefficients.
• A vector bundle is orientable iff regarding real K- theory. it has a spin structure, so if the first and second Stiefel -Whitney class disappear.
• A vector bundle is orientable iff regarding complex K- theory. it has a spinc structure.

A cohomological orientation of a manifold is by definition a cohomological orientation of its tangent bundle. Milnor -Spanier duality provides a bijection between homological and cohomological orientations of a closed manifold with respect to a given ring spectrum.