Encyclopedia
In
mathematics, a
vector bundle is a geometrical construct where to every point of a topological space we attach a vector space in a compatible way, so that all those vector spaces, "glued together", form another topological space . A typical example is the tangent bundle of a
differentiable manifold: to every point of the manifold we attach the tangent space of the manifold at that point. Or consider a smooth curve in
R2, and attach to every point of the curve the line normal to the curve at that point; this yields the "normal bundle" of the curve.
This article deals mostly with
real vector bundles, with finite-dimensional fibers.
Complex vector bundles are important in many cases, too; they are a special case, meaning that they can be seen as extra structure on an underlying real bundle.
Definition and first consequences
A real vector bundle is given by the following data:
- topological spaces X and E
- a continuous map π : E → X
- for every x in X, the structure of a real vector space on the fiber π−1
satisfying the following compatibility condition: for every point in
X there is an open neighborhood
U, a natural number
n, and a homeomorphism φ :
U ×
Rn → π
−1 such that for every point
x in
U:
- πφ = x for all vectors v in Rn
- the map v φ yields an isomorphism between the vector spaces Rn and π−1.
The open neighborhood
U together with the homeomorphism φ is called a
local trivialization of the bundle. The local trivialization shows that "locally" the map π looks like the projection of
U ×
Rn on
U.
A vector bundle is called
trivial if there is a "global trivialization", i.e. if it looks like the projection
X ×
Rn →
X.
Every vector bundle π :
E →
X is
surjective, since vector spaces cannot be empty.
Every fiber π
−1 is a finite-dimensional real vector space and hence has a dimension
dx. The function
x dx is locally constant, i.e. it is constant on all connected components of
X. If it is constant globally on
X, we call this dimension the
rank of the vector bundle. Vector bundles of rank 1 are called line bundles.
Vector bundle morphisms
A
morphism from the vector bundle π
1 :
E1 →
X1 to the vector bundle π
2 :
E2 →
X2 is given by a pair of continuous maps
f :
E1 →
E2 and
g :
X1 →
X2 such that
- for every x in X1, the map π1−1 → π2−1 induced by f is a linear transformation between vector spaces.
The class of all vector bundles together with bundle morphisms forms a category. Restricting to smooth manifolds and smooth bundle morphisms we obtain the category of smooth vector bundles.
We can also consider the category of all vector bundles over a fixed base space
X. As morphisms in this category we take those morphisms of vector bundles whose map on the base space is the identity map on
X. That is, bundle morphisms for which the following diagram
commutes:
Sections and locally free sheaves
Given a vector bundle π :
E →
X and an open subset
U of
X, we can consider
sections of π on
U, i.e. continuous functions
s :
U →
E with π
s = id
U. Essentially, a section assigns to every point of
U a vector from the attached vector space, in a continuous manner.
As an example, sections of the tangent bundle of a differential manifold are nothing but
vector fields on that manifold.
Let
F be the set of all sections on
U.
F always contains at least one element, namely the
zero section: the function
s that maps every element
x of
U to the zero element of the vector space π
−1. With the pointwise addition and scalar multiplication of sections,
F becomes itself a real vector space. The collection of these vector spaces is a sheaf of vector spaces on
X.
If
s is an element of
F and α :
U →
R is a continuous map, then α
s is in
F. We see that
F is a module over the ring of continuous real-valued functions on
U. Furthermore, if O
X denotes the structure sheaf of continuous real-valued functions on
X, then
F becomes a sheaf of O
X-modules.
Not every sheaf of O
X-modules arises in this fashion from a vector bundle: only the locally free ones do.
Even more: the category of real vector bundles on
X is
equivalent to the category of locally free and finitely generated sheaves of O
X-modules.
So we can think of the vector bundles as sitting inside the category of sheaves of O
X-modules; this latter category is abelian, so this is where we can compute kernels of morphisms of vector bundles.
Operations on vector bundles
Two vector bundles on
X, over the same field, have a
Whitney sum, with fibre at any point the direct sum of fibres. In a similar way,
fibrewise tensor product and dual space bundles may be introduced.
Variants and generalizations
Vector bundles are special
fiber bundles, loosely speaking those where the fibers are vector spaces.
Smooth vector bundles are defined by requiring that
E and
X be
smooth manifolds, π :
E →
X be a smooth map, and the local trivialization maps φ be diffeomorphisms.
Replacing real vector spaces with
complex ones, we obtain complex vector bundles. This is a special case of reduction of the structure group of a bundle. Vector spaces over other topological fields may also be used, but that is comparatively rare.
If we allow arbitrary Banach spaces in the local trivialization , we obtain
Banach bundles.
K-theory
The K theory groups K of a manifold are defined as the abelian group generated by isomorphism classes [E] of vector bundles modulo the relation that whenever we have an exact sequence 0->A->B->C->0 then [B]=[A]+[C] in K-theory. KO-theory is a version of this construction which considers real vector bundles. K-theory with compact supports can also be defined, as well as higher K-theory groups.
The famous periodicity theorem of Raoul Bott asserts that the K theory of any space X is isomorphic to that of X x two dimensional sphere.
In algebraic geometry, one considers the K theory groups consisting of coherent sheaves on a scheme X, as well as the K theory groups of vector bundles on the scheme with the above equivalence relation. The two constructs are the same provided that the underlying scheme is smooth.
References
- Jurgen Jost, Riemannian Geometry and Geometric Analysis, Springer-Verlag, Berlin ISBN 3-540-4267-2 See section 1.5.
- Ralph Abraham and Jarrold E. Marsden, Foundations of Mechanics, Benjamin-Cummings, London ISBN 0-8053-0102-X See section 1.5.
