Orientation (mathematics)

The orientation is a term used in linear algebra and differential geometry . In a -dimensional space, two ordered bases have the same orientation if they emerge from each other through linear mappings with positive determinants of the mapping matrix ( e.g. stretching and rotating ). If reflections are also required, the determinant is negative and the bases are not oriented in the same way. ${\displaystyle n}$

There are two possible orientations, a change between the orientations is not possible by turning. Illustrative examples:

• One-dimensional:
  • Reading direction of character strings (see also palindromes ) or single-stranded nucleic acids
• In the plane:
  • Mirror writing has a different orientation than writing.
  • Clocks rotate clockwise and not counterclockwise.
• In the room:
  • My reflection has a different orientation than me.
  • Screws with right hand threads have an orientation other 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 have no spatial depth.

Orientation of a vector space

Definitions

Let be a finite-dimensional vector space with two ordered bases and . There is also a base change matrix that describes the transition from one base to the other. If and is more precise , they can be represented as linear combinations with regard to the basis . is then the matrix formed from the . As a basic change matrix, this is always bijective and therefore has a determinant different from 0 , that is, it is or . The determinant is positive, it is said that the bases and have the same orientation . The change of base itself is called orientation- preserving with a positive determinant , otherwise orientation- reversing . Since use was made here of the arrangement of real numbers, this definition cannot be transferred to vector spaces over arbitrary bodies , but only to those over ordered bodies . ${\displaystyle V}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle T_{B}^{A}}$ ${\displaystyle A=(a_{1},\dotsc ,a_{n})}$ ${\displaystyle B=(b_{1},\dotsc ,b_{n})}$ ${\displaystyle a_{j}}$ ${\displaystyle B}$ ${\displaystyle a_{j}=\lambda _{1,j}b_{1}+\dotsb +\lambda _{n,j}b_{n}}$ ${\displaystyle T_{B}^{A}}$ ${\displaystyle \lambda _{i,j}}$ ${\displaystyle \det(T_{B}^{A})>0}$ ${\displaystyle \det(T_{B}^{A})<0}$ ${\displaystyle A}$ ${\displaystyle B}$

The orientation is an equivalence relation of a intermediate bases - vector space defined. Two bases and are equivalent if they have the same orientation. There are two equivalence classes with regard to this equivalence relation. The determinant multiplication theorem and the fact that base transformations are reversible ensures that this equivalence relation is well-defined and that there are actually only two equivalence classes . Each of these two equivalence classes is now called an orientation . An orientation of a vector space is thus indicated by specifying an equivalence class of bases, for example by specifying a base belonging to this equivalence class. Each base belonging to the selected equivalence class is then called positively oriented , the others are called negatively oriented . ${\displaystyle \mathbb {R} }$ ${\displaystyle A}$ ${\displaystyle B}$

example

In are both , and ordered bases. The base transformation matrix is ​​thus ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle (e_{1},e_{2})}$ ${\displaystyle (e_{2},e_{1})}$

${\displaystyle M=\left({\begin{matrix}0&1\\1&0\end{matrix}}\right)}$.

The determinant of is . So the two bases are not oriented in the same way and are representatives of the two different equivalence classes. ${\displaystyle M}$ ${\displaystyle \det(M)=-1}$

This can be easily illustrated: The first base corresponds to a “normal” coordinate system in which the axis “points” to the right and the axis upwards. If you swap the two axes, ie the -axis "points" upwards and the -axis to the right, you get a second base with a different orientation. ${\displaystyle (x,y)}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle y}$

Similarly, in the three-dimensional visual space (with a fixed coordinate system), one can speak of right and left systems , which can be differentiated using the three-finger rule .

Homological and cohomological orientation

With a real will continue -dimensional vector space designated with the relative homology of the space pair . In homology theory it has been shown that an isomorphism exists. Choosing an orientation for therefore corresponds to choosing one of the two producers of . ${\displaystyle V}$ ${\displaystyle n}$ ${\displaystyle H_{n}(V,V\setminus 0;\mathbb {Z} )}$ ${\displaystyle (V,V\setminus 0)}$ ${\displaystyle H_{n}(V,V\setminus 0;\mathbb {Z} )\simeq \mathbb {Z} }$ ${\displaystyle V}$ ${\displaystyle H_{n}(V,V\setminus 0;\mathbb {Z} )}$

For this, one looks at an embedding of the -dimensional standard simplex , which maps the barycenter according to (and consequently the side surfaces according to ). Such a map is a relative cycle and represents a producer of . Two such embeddings represent the same generator if they are both orientation-preserving or both are not orientation-preserving. ${\displaystyle n}$ ${\displaystyle V}$ ${\displaystyle 0}$ ${\displaystyle V\setminus 0}$ ${\displaystyle H_{n}(V,V\setminus 0;\mathbb {Z} )}$

Because dual is to, an orientation and the associated choice of a producer of also defines a producer of . ${\displaystyle H^{n}(V,V\setminus 0;\mathbb {Z} )}$ ${\displaystyle H_{n}(V,V\setminus 0;\mathbb {Z} )}$ ${\displaystyle H_{n}(V,V\setminus 0;\mathbb {Z} )}$ ${\displaystyle H^{n}(V,V\setminus 0;\mathbb {Z} )}$

Orientation of a manifold

A non-orientable manifold - the Möbius strip

Definition (using the tangent space)

An orientation of a -dimensional differentiable manifold is a family of orientations for each individual tangential space , which continuously depends on the base point in the following sense : ${\displaystyle {\mathcal {O}}=\left\{{\mathcal {O}}_{p}\right\}_{p\in M}}$ ${\displaystyle n}$ ${\displaystyle M}$ ${\displaystyle {\mathcal {O}}_{p}}$ ${\displaystyle T_{p}M}$ ${\displaystyle p}$

For each point there is a map with coordinate functions , ..., defined on an open neighborhood of , so that at each point the basis induced by the map in the tangent space ${\displaystyle p\in M}$ ${\displaystyle U}$ ${\displaystyle p}$ ${\displaystyle \varphi \colon U\to V\subset \mathbb {R} ^{n}}$ ${\displaystyle x^{1}\colon U\to \mathbb {R} }$ ${\displaystyle x^{n}\colon U\to \mathbb {R} }$ ${\displaystyle q\in U}$ ${\displaystyle T_{q}M}$

${\displaystyle \left(\left.{\frac {\partial }{\partial x^{1}}}\right|_{q},\dotsc ,\left.{\frac {\partial }{\partial x^{n}}}\right|_{q}\right)}$

is positively oriented. ${\displaystyle {\mathcal {O}}_{q}}$

A manifold is orientable if such an orientation exists. The following theorem provides an equivalent characterization of orientability:

${\displaystyle M}$is orientable if and only if an atlas of exists, so that applies to all maps with non-empty intersection and to all in the domain of : ${\displaystyle {\mathcal {A}}}$ ${\displaystyle M}$ ${\displaystyle \varphi ,\psi }$ ${\displaystyle U^{\varphi }\cap U^{\psi }\neq \emptyset }$ ${\displaystyle x}$ ${\displaystyle \psi (U^{\varphi }\cap U^{\psi })}$ ${\displaystyle \varphi \circ \psi ^{-1}}$

${\displaystyle \det \left(D_{x}(\varphi \circ \psi ^{-1})\right)>0}$

Here denotes the Jacobi matrix . ${\displaystyle D_{x}}$

Coordinate-free definition

Let be a smooth, -dimensional manifold. This manifold is orientable if and only if it exists on a smooth, non-degenerate - form . ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle \alpha }$

Homological orientation of a manifold

Let be a -dimensional (topological) manifold and a ring. With the help of the cutting axiom for a homology theory one obtains: ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle R}$

${\displaystyle H_{n}(M,M\setminus \{x\})\cong H_{n}(\mathbb {R} ^{n},\mathbb {R} ^{n}\setminus \{0\})\cong R}$

An orientation towards is a selection of producers ${\displaystyle R}$ ${\displaystyle M}$

${\displaystyle \{\mu _{x}\in H_{n}(M,M\setminus \{x\})|x\in M\}}$

with the following compatibility condition: For each there is an open environment and an element , so that for all the mapping induced by the inclusion of space pairs on the homology ${\displaystyle x\in M}$ ${\displaystyle U\subset M}$ ${\displaystyle \mu _{U}\in H_{n}(M,M\setminus U)}$ ${\displaystyle y\in U}$

${\displaystyle H_{n}(M,M\setminus U)\to H_{n}(M,M\setminus \{y\})}$

the element to mapping. For example, the concept of orientation coincides with the usual concept of orientation. However, other results can be obtained for other rings; for example every manifold is -orientable. ${\displaystyle \mu _{U}}$ ${\displaystyle \mu _{y}}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} _{2}}$

Generalized theories of homology

Let be a (reduced) generalized homology theory given by a ring spectrum . We denote by the image from below the iterated suspension isomorphism . For a closed -manifold , a point and an open neighborhood, let us be a continuous map that is a homeomorphism on and constant on the complement of . Then a homology class is called ${\displaystyle {\tilde {h}}_{*}}$ ${\displaystyle s_{n}\in {\tilde {h}}_{n}(S^{n})}$ ${\displaystyle 1\in {\tilde {h}}_{0}(S^{0})}$ ${\displaystyle n}$ ${\displaystyle M}$ ${\displaystyle m\in M}$ ${\displaystyle m\in U\simeq D^{n}}$ ${\displaystyle \epsilon ^{m,U}\colon M\to S^{n}}$ ${\displaystyle U}$ ${\displaystyle U}$

${\displaystyle [M]\in {\tilde {h}}_{n}(M)}$

an orientation or fundamental class if ${\displaystyle h}$ ${\displaystyle h}$

${\displaystyle \epsilon _{*}^{m,U}[M]=\pm s_{n}}$

applies to all . For the singular homology , this definition agrees with the above. ${\displaystyle m,U}$

Orientation of a vector bundle

An orientation of a vector bundle is a family of orientations for each individual fiber that is continuously dependent on the base point in the following sense : ${\displaystyle {\mathcal {O}}=\left\{{\mathcal {O}}_{b}\right\}_{b\in B}}$ ${\displaystyle p\colon E\to B}$ ${\displaystyle {\mathcal {O}}_{b}}$ ${\displaystyle F_{b}=p^{-1}(b)}$ ${\displaystyle b\in B}$

To each point there is an open environment by using local Trivialisation , such that for each by ${\displaystyle b\in B}$ ${\displaystyle U}$ ${\displaystyle b}$ ${\displaystyle h\colon U\times \mathbb {R} ^{n}\to p^{-1}(U)}$ ${\displaystyle b\in B}$

${\displaystyle x\mapsto h(b,x)}$

defined mapping from to is orientation preserving . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle F_{b}}$

A manifold is then orientable if and only if its tangential bundle is orientable.

Cohomological formulation: For an orientable -dimensional vector bundle with zero cut applies for and there is a generator of , the restriction of which corresponds to the selected orientation of the fiber for each . ${\displaystyle n}$ ${\displaystyle E\to B}$ ${\displaystyle E_{0}}$ ${\displaystyle H^{i}(E,E-E_{0};\mathbb {Z} )=0}$ ${\displaystyle 0<i<n}$ ${\displaystyle u\in H^{n}(E,E-E_{0};\mathbb {Z} )\simeq \mathbb {Z} }$ ${\displaystyle H^{n}(F_{b},F_{b}-0;\mathbb {Z} )}$ ${\displaystyle b\in B}$ ${\displaystyle F_{b}}$

The cohomology class corresponding to a chosen orientation

${\displaystyle u\in H^{n}(E,E-E_{0};\mathbb {Z} )}$

is called the Thom class or orientation class of the oriented vector bundle.

Alternatively, one can also use the Thom space , the cohomology of which is too isomorphic. The Thom class then corresponds to the image of the neutral element (with regard to the cup product ) under the Thom isomorphism . ${\displaystyle Th(E)}$ ${\displaystyle H^{*}(Th(E);\mathbb {Z} )}$ ${\displaystyle H^{*}(E,E-E_{0};\mathbb {Z} )}$ ${\displaystyle 1\in H^{0}(B;\mathbb {Z} )}$ ${\displaystyle H^{*+n}(Th(E);\mathbb {Z} )\simeq H^{*}(B;\mathbb {Z} )}$

Cohomological Orientation (Generalized Cohomology Theories)

Let be a (reduced) generalized cohomology theory with a neutral element given by a ring spectrum . We denote by the image from below the iterated suspension isomorphism . For each , inclusion induces a mapping . A cohomological orientation with regard to the cohomology theory is - by definition - an element ${\displaystyle {\tilde {h}}^{*}}$ ${\displaystyle 1\in {\tilde {h}}^{0}(S^{0})}$ ${\displaystyle \sigma ^{n}\in {\tilde {h}}^{n}(S^{n})}$ ${\displaystyle 1}$ ${\displaystyle b\in B}$ ${\displaystyle \mathbb {R} ^{n}=F_{b}\to E}$ ${\displaystyle j_{b}\colon S^{n}\to Th(E)}$ ${\displaystyle {\tilde {h}}}$

${\displaystyle u\in {\tilde {h}}^{n}(Th(E))}$

with for everyone . ${\displaystyle j_{b}^{*}u=\pm \sigma ^{n}}$ ${\displaystyle b\in B}$

Examples:

• In the case of singular cohomology with coefficients, this corresponds to the definition above and is the Thom class. ${\displaystyle \mathbb {Z} }$ ${\displaystyle {\tilde {h}}^{*}(X)={\tilde {H}}^{*}(X;\mathbb {Z} )}$ ${\displaystyle u}$
• Each vector bundle can be oriented with respect to singular cohomology with coefficients. ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$
• A vector bundle can be orientated with respect to real K-theory if and only if it has a spin structure , i.e. if the first and second Stiefel-Whitney classes vanish.
• A vector bundle can be oriented with regard to complex K-theory if and only if it has a spin ^C structure.

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

literature

• Gerd Fischer : Linear Algebra. 14th revised edition. Vieweg-Verlag, Wiesbaden 2003, ISBN 3-528-03217-0 .
• Ralph Abraham, Jerrold E. Marsden , Tudor Ratiu: Manifolds, Tensor Analysis, and Applications (= Applied Mathematical Sciences 75). 2nd edition. Springer-Verlag, New York et al. 1988, ISBN 0-387-96790-7 (also reprint of this edition: 2009).
• Klaus Jänich : Vector analysis. 2nd Edition. Springer-Verlag, Berlin et al. 1993, ISBN 3-540-57142-6 , pp. 70ff.

Web links

• Orientation of manifolds (Manifold Atlas)
• Orientation of manifolds in generalized cohomology theories (Manifold Atlas)

Individual evidence

1. ^ Allen Hatcher: Algebraic Topology . University Press, Cambridge 2000, ISBN 0-521-79540-0 , pp. 231 ( online ).