Cohomology and vector bundles corrin clarkson reu 2008 september 28, 2008 abstract vector bundles are a generalization of the cross product of a topological space with a vector space. M s1 which we claim is a non trivial real vector bundle of rank 1. A vector bundle of rank 1 is usually called a line bundle. Bring vector space back in picture cliff construct manifolds from given manifolds understand. Vector bundles of rank 1 is also called the line bundle. An example of vector bundle which is not trivial but nevertheless stably trivial is ts2 definition 10.
While we accomplish this in a certain sense by showing that any vector bundle e. If the dimension of the vector space is mthen the bundle is often called an mplane bundle. In this study, we illustrate an application of vector bundles by prolonging isometric actions from a manifold base manifold to another by using a vector bundle structure. E intersecting each ber of e in a vector subspace, such that the restriction pj e 0 is a. M s1 which we claim is a non trivial real vector bundle of rank 1 over s1. Applying this operation on each bre of a vector bundle e. If mis compact, then for any vector bundle eover m, there exists a vector bundle f over mso that e f is a trivial bundle over m. The elements of the fiber over x in a vector bundle can be viewed as abstract vectors in an internal space, with the local trivialization. Often the vector bundles that one encounters do not arise right away as in the. A rank n vector bundle on a manifold m is a bration. Luckily, there is again a simpler criterion which ensures integrability. Suppose we have a vector bundle on m, along with a trivialization over an open cover ui. Im not sure if your comment is in response to mine. We have put a topology on fesuch that for each local trivialization.
The most important example is the tangent bundle, a real vector bundle. M are vector spaces, the manifolds f in the local trivialization are vector spaces, and for each p. The notion of vector bundle is a basic extension to the geometric domain of the fundamental idea of a vector space. They are referred as rank 2 stable vector bundles on the complex projective space p3 8. We have put a topology on fesuch that for each local trivialization ej u u rn as.
A smooth real vector bundle of rank k over the base manifold m is a manifold e. An isomorphism of a rank k vector bundle e over x with the trivial bundle of rank k over x is called a trivialization of e, and e is then said to be trivial or trivializable. Bundles, classifying spaces and characteristic classes contents. B, onto b, is called a vector bundle of rank kover bif the following holds. Vectors v x2p x are thought of as arrows with their tail at x. The local trivialization shows that locally the map. For instance, to any vector space v we can associate its dual space v. We have developed the notion of a cp vector bundle over xas a certain kind of cp mapping e xthat is roughly speaking a cp varying family of. A vector bundle for which a trivialization exists is called trivializable accordingly, the local triviality condition in the definition of topological vector bundles def.
Prolongations of isometric actions to vector bundles. A group acting on it is found and it is shown that it acts in. Bis isomorphic to the pullback of a universal vector bundle e0. We have seen that eis a vector sub bundle of a trivial bundle m rn over m. Mequipped with a connection such that there exists a trivialization in which the covariant derivative has the form above over each open set in the trivialization. By definition, a smooth kvector bundle of rank r over a smooth manifold m. For any smooth manifold m, e m rk is a trivial bundle over m. Subbundles and quotient bundles motivation example n. The euler class, the thom class and the chern classes are a few. U r ntaking p b to b r by a vector space isomorphism for every b2u. In the case there is no ambiguity about the base, we will denote a vector bundle by efor short.
Cohomology and vector bundles university of chicago. The vector bundle of rank rover xis said to be trivial if there exists a global bundle chart e x. A vector bundle is said to be stably trivial if it becomes trivial after taking the direct sum with a vector bundle isomorphic to the trivial bundle. Rnbe a trivialization, and denote e i to be the ith vector in the standard basis of rn. If the original data of clutching functions came from locally trivializations of a bundle. Induced bundles and vector bundles in the previous lecture, we have seen how principal gbundles correspond to equivalence classes of cocycles. U ck that is a biholomorphic map of complex manifolds. One can recover the ber bundle from this principal bundle by a simple construction of. Given a space x, we take a real or complex finite dimensional vector space v and.
Im using the letter m to denote the base space of the vector bundle as a concession to the fact that in most of the applications well be. Vector bundles in this section we recall some basics on vector bundles. We prefer to think of sections as \vector valued maps on m, which can be added and multiplied by scalars, and wed like to think of the directional derivative dsxx as something which respects this linear structure. From this perspective the answer is ambiguous, unless e happens to. Vector bundle examples trivial bundle moebius bundle tangent bundle cotangent bundle. This is a 2 dimensional vector space with the point xas its zero vector 0 x. Motivation let x,o be a cp premanifold with corners, 0. Linear algebra operations on vector bundles motivation. Characteristic classes assign to the each vector bundle a cohomology class of the base space. Aug 11, 2019 an isomorphism of a rank k vector bundle e over x with the trivial bundle of rank k over x is called a trivialization of e, and e is then said to be trivial or trivializable. M is a complex vector bundle together with the structure of a complex manifold on e, such that for any x2mthere exists u3x in mand trivialization. We can also consider the category of all vector bundles over a fixed base space x. Fiber bundles, especially vector bundles, are ubiquitous in mathematics. E m is a choice of orientation for every fiber ex, such that for any local trivialization.
For the trivialization neighborhoods, only one is needed. A simple example of a non trivial vector bundle is the in. Any example of manifold without global trivialization of. In mathematics, an orientation of a real vector bundle is a generalization of an orientation of a vector space.
Cn yang, 1979 in this lecture we introduce the notion of a principal. Any example of manifold without global trivialization of tangent bundle. Vector bundles and pullbacks nikolai nowaczyk november 2008 contents. The definition of a vector bundle shows that any vector bundle is locally trivial. This construction can also be applied brewise to vector bundles. The trivial vector bundle of rank r over m is the product m. In this case x is also a section of the vector bundle. But there are many other interesting vector bundles. We say that a vector bundle is locally trivial, if for every x2xthere is an open neighborhood x2u. Holomorphic vector bundles northwestern university. E f is the tensor product bundle, which has rank rk erk f. Accordingly, the local triviality condition in the definition of topological vector bundles def. Its local coordinate charts are obtained by restricting the trivialization.
Wellknown examples of vector bundles are the tangent and cotangent bundles, tmand t m, as well as the tensor bundle tm x2m p. The tangent bundle tm is indeed a vector bundle, of rank dimm. This new vector bundle fe is called the associated vector bundle to e, given by the functor f. We have seen that eis a vector subbundle of a trivial bundle m rn over m. Several other approaches of vector bundles can be found in 35, 15. Intuitively this is the object we get by gluing at each point p. Ebe a cp vector bundle map between cp vector bundles over a cp premanifold with corners x.
Vector bundles and charts if you open any book on di. Vector bundles and connections universiteit utrecht. U is called a local trivialization of e over u and u is a trivializing neighbour hood for. In order to help motivate them, we rst discuss vector bundles, and how they. A vector bundle for which a trivialization exists is called trivializable. As one learns in an introductory smooth manifolds course, for any smooth man ifold m there is a naturally associated vector bundle. We will call e the total space, m the base, 1p the fiber over p, and.
For this goal, we study the moduli space of 5tuples x,x,z,e. On a connected component of xall bers of a locally trivial vector bundle have the same. It is a real vector bundle of rank n dimm which in general is not trivial. Now we put an inner product on rn, and take the ber f p of f at p2mto be the orthogonal complement of e. The open neighborhood u together with the homeomorphism is called a local trivialization of the vector bundle. Using a partition of unity argument assuming youre working over a sufficiently nice space, or your vector bundle is a pullback of a universal one look up numerable cover for more on this but note that all the answers to this question tacitly assume this, any finite dimensional vector bundle admits a continuous choice of inner product. Orientations on vector bundles universiteit utrecht. A bundle over a manifold is trivial if it is simply the cartesian product of. Property t of vector bundles is in fact stronger than it appears to be. Vector bundles and structure groups a vector bundle over a topological space m or with base space m is, essentially, family of vector spaces continuously parametrized by m. As we have seen in the previous section, holomorphic vector bundles are complex vector bundles for which the local trivialization has been chosen in such a way that the transition functions are holomorphic.
1384 1629 752 545 1079 1303 398 1668 780 551 966 633 42 1138 879 870 723 1591 1206 634 1047 191 1418 1082 207 1238 731 667 1409 153