Vector bundle
Vector bundles (or sometimes detailed vector bundles ) are families of vector spaces parametrized by the points of a topological space. The concept of the basis of a vector space can be generalized to such specific fiber bundle and is called frame.
Clearly, there is a vector bundle of one vector space for each point of the base space. As vector spaces of the same dimension but are always isomorphic, the essential information lies in the relations between these vector spaces. The best-known example of a vector bundle is the tangent bundle of a differentiable manifold. The relationship between the different tangent spaces, ie, the vector spaces for each point, expressed for example in the question of whether a vector field is differentiable.
The question of how vector bundles can look to a room is closely correlated with global topological properties of the space. Non- isomorphic vector bundles can often be distinguished by their characteristic classes.
- 3.1 homomorphism
- 3.2 isomorphism
- 3.3 Example
- 4.1 sub-vector bundle
- 4.2 Restricted vector bundles
- 5.1 Interface
- 5.2 framework
- 6.1 differentiable vector bundle
- 6.2 holomorphic vector bundle
- 6.3 G- vector bundle
- 9.1 Definition
- 9.2 Local Free sheaf
- 9.3 Locally Free Sheaves and Vector bundles
Definitions
Vector bundles
Be a real or complex n- dimensional vector space. A real or complex vector bundle of rank is a triple consisting of topological spaces ( total space ) and ( base ) and a continuous surjective map, such that:
- For each point of the fiber by transferring the structure of a real or complex N- dimensional vector space.
- " Local triviality ": For each point there exists a neighborhood of and a homeomorphism
A vector bundle is called trivial if there is a trivialization with. is a trivial vector bundle.
Line bundle
A vector bundle of rank 1 is called a line bundle ( as a mistranslation from English also line bundle).
Examples
- The tangent bundle of a differentiable manifold is a vector bundle consisting of the tangent spaces of the manifold. Correspondingly, the cotangent bundle consisting of the Kotangentialräumen - ie the dual space of the tangent space - a vector bundle.
- The Möbius strip is a line bundle over the 1- sphere (circle). Locally it is homeomorphic with an open subset of is. However, the Möbius strip is not homeomorphic to what would be a cylinder.
- The space of differential forms is a bundle of the outer algebra also a vector bundle.
- The - Tensorbündel is also a vector bundle, which includes the previously listed vector bundles as special cases.
Homomorphism of vector bundles
Homomorphism
A Vektorbündelhomomorphismus of the vector bundle in the vector bundle is a pair of continuous maps and so
- Applies and
- For all is a linear map.
Often a Vektorbündelhomomorphismus is briefly referred to as Bündelhomomorphismus or as a homomorphism.
Isomorphism
A Vektorbündelhomomorphismus of after is a Vektorbündelisomorphismus, if and are homeomorphisms and the induced linear map is a Vektorraumisomorphismus.
Example
Looking at the circle as a manifold, then the tangent bundle of isomorphic to the trivial vector bundle. The homeomorphism between the base spaces is the identity map and reads between the total spaces
For and.
Substructures
Sub-vector bundle
With the fibers of the vector bundle are referred to at point. A sub-vector bundle of the vector bundle is consisting of a topological subspace of a family of vector subspaces of, so that a separate vector bundle.
Restricted vector bundles
With the fibers of the vector bundle are again referred to at point and called a topological subspace. The product subject to restricted vector bundle is defined by
The restricted vector bundle is a standalone vector bundle with respect to the topological sub- space.
Other objects in vector bundles
Section
If U is an open subset of B, it means a picture
For which holds a section of E through U. The quantity Γ (U, E ) of all sections of E through U is a vector space.
Framework
Under a frame ( Frame in English ) refers to a kind of basis of a vector bundle. Is a subset of the vector beam forming at each point of a base of the corresponding vector space. Precisely, this means:
Be a vector bundle of rank and be an open subset of the base space. A local frame of over is an ordered n- tuple. This is for all i is a cut in on, so that forms a vector space basis of the fiber for all. If you can choose, then one speaks of a global framework.
Vector bundles with additional structures
Differentiable vector bundle
Be a vector bundle. Are and differentiable manifolds and the projection and the trivialization differentiable, this means differentiable vector bundles. It is called smooth if the manifolds are smooth and the pictures are infinitely differentiable.
Holomorphic vector bundles
A holomorphic vector bundle is a complex vector bundle over a complex manifold, so that the total space of a complex manifold, and the projection is a holomorphic map.
G- vector bundles
Let be a group. If and G-spaces, then a vector bundle is a G - vector bundle if the group effect
For all is a linear map.
Classifying space and classifying map
The classifying space of - dimensional real vector bundle is the Grassmann manifold of dimensional subspaces in, this is referred to as. This means that every - dimensional real vector bundle is of the form of a continuous map (called the classifying map of the bundle ) and the tautological bundle and two bundles are isomorphic if and only if their classifying pictures are homotopic.
Analogously, the Grassmann manifold of dimensional subspaces in, the classifying space for - dimensional complex vector bundle.
Stable vector bundles
Two vector bundles and hot stably equivalent if there trivial vector bundle (not necessarily the same dimension ) with
There. The equivalence classes of this equivalence relation are called stable vector bundles. ( This definition has no relation with the notion of stable vector bundles in algebraic geometry. )
Let and the ascending associations (ie the Kolimiten regarding the means defined inclusions and ), then you can consider a vector bundle and its classifying image or the composition with the inclusion or. Two vector bundles are stable if and only equivalent if the corresponding figures are respectively homotopic.
Vector bundles in algebraic geometry
Definition
For ( algebraic ) vector bundles in algebraic geometry and schemes for all points of a vector space, and the local trivializations are isomorphisms
Mostly, however, is meant (see below) with " vector bundles " in algebraic geometry, a locally free sheaf.
Local free sheaf
Let ( X, OX) is a locally ringed space, for example, a topological space with the sheaf of continuous real - or complex-valued functions, a differentiable manifold with the sheaf of C ∞ - functions or a schema.
A locally free sheaf is an OX - module M which is locally isomorphic to a free OX- module, ie X can be covered by open sets U such that M | U is isomorphic to a direct sum of copies of OX | U.
Local free sheaves and vector bundles
The following two constructions provide in the case of topological spaces or differentiable manifolds is an equivalence of categories of locally free sheaves and vector bundles on X (for simplicity's sake the notation is the case of real vector bundles over a topological space described ):
- A vector bundle is associated with the sheaf of its sections.
- A locally free sheaf M is assigned to the disjoint union of its fibers E Mx / mxMx. We choose an open cover (Ui ) of X such that M on each Ui is trivial. A trivialization defined nowhere vanishing n -sections of M over U, the fibers, form a base. These define a mapping
In the case of algebraic geometry, this construction is somewhat simpler: the bundle to a locally free sheaf E is
While S denotes the symmetric algebra and the Spec Algebrenspektrum.
Further terms
- The investigation into the use of stable equivalence classes of vector bundles is the subject of K- theory.
- On algebraic curves (semi-) stable vector bundles have particularly good properties.