A section of a fiber bundle is a function whose output range is continuously dependent on the input. In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space x for example x could be a topological space, a manifold, or an algebraic variety. We we say that two vector bundles over m are isomorphic if there is a bundle map. This homotopy fiber may be thought of as the internal principal. In order to help motivate them, we rst discuss vector bundles, and how they. We will consider rst the simplest theory of vector bundles where the vector space is a onedimensional complex vector space line bundles. This seminar is an account of alan weinsteins recent paper the volume of a differentiable stack. Then 1 there is an ainvariant, pointwise decomposition of v. B, onto b, is called a vector bundle of rank kover bif the following holds. Vector bundles are almost always required to be locally trivial, however, which means they are examples of fiber bundles. Specifically, under mild technical assumptions a fiber bundle always has the homotopy lifting property or homotopy covering property see steenrod 1951, 11. Finite group actions on moduli spaces of vector bundles arxiv. 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. Let v be a real or complex vector space, and let glv be the group of.
There are no triple overlaps, so the cocycle condition is vacuously satis. The upshot of this is that ndimensional vector bundles are classi ed by nonabelian cohomology h1x. In particular, we show that for trivial bundles the oka principle is valid. So the sheaf of groups in question will be that of germs of continuous gln, avalued. The transition function satisfy the cocycle condition. A compact complex manifold m is called a hodge manifold if there exists a positive line bundle l over m. Nevertheless, since p is locally trivial, local sections do exist. From this it follows that there is a canonical bijective correspondence between isomorphism classes of vector bundles of rank kon mand equivalence classes of glkcocycles on m.
Dual of a holomorphic vector bundle mathematics stack exchange. The subtle and vital point is that it is possible to create quite different bundles which nonetheless look similar in small patches. In this paper we take f to be a transitive anosov di. Finite group actions on moduli spaces of vector bundles florent schaffhauser abstract. An isomorphism is a morphism of line bundles that admits an inverse. Intuitively this is the object we get by gluing at each point p. Rd, a linear cocycle can be represented by a matrixvalued function a. Vector bundles in algebraic geometry enrique arrondo notes prepared for the first summer school on complex geometry villarrica, chile 79 december 2010 1. The cocycle construction of dual bundle is easy for definition but it does not lie in my head solidly. A bundle over a manifold is trivial if it is simply the cartesian product of.
H x, a \omega \in \mathbfhx,a a cocycle, that the object classified by the cocycle is its homotopy fiber p x p \to x regarded as an object in the overcategory over x x. Almost synonymous terms used in various areas are topological bundle, locally trivial fibre bundle, fibre space, fibration, skew product etc. The last condition is known as the cech cocycle condition more. In mathematics, a vector bundle is a topological construction that makes precise the idea of a. Other kinds of space may be attached, thus entering the more general theory of fibre bundles. Conversely, any holomorphic matrix cocycle inscribed in an open covering of t determines a holomorphic vector bundle over t. In general, a section of a map fis a right inverse of for, in other words, a map g going the other way f g such that fg id. Induced bundles and vector bundles in the previous lecture, we have seen how principal gbundles correspond to equivalence classes of cocycles.
If the dimension of the vector space is mthen the bundle is often called an mplane bundle. Vector bundles, connections and curvature 3 there is an obvious projection t. Ill explain that differentiable stacks are a generalization of smooth manifolds and that they crop up in many interesting situations, like the study of. A basis for this infinite dimensional vector space, endowed with the frechet topology of uniform convergence of maps and each of their derivatives is the denumerable set we suppress the notation.
In the first 25 pages he describes bundles as given by a cocycle of coordinate transformations and shows that a continuous map between automorphism groups defines a new bundle, he calls a tensor bundle. Bring vector space back in picture cliff construct manifolds from given manifolds understand manifolds i. Induced bundles and vector bundles induced bundles. It is interesting to know, how far we can generalize the notion of a groupvalued cocycle keeping the property to determine a bundle. Insection3,weshall be interested in the much more involved classi. A vector bundle with fiber v is a triple e, b, p where e and b are smooth. The isomorphism class of a vector bundle constructed from a glk cocycle depends only on the equivalence class of the cocycle. S2 of rank 2 for each k2z or a complex vector bundle of rank 1, since g 01. Vector bundles and connections the exposition of vector. Thus there exists an open cover of u fugso that mj u is trivial when restrict to u2u.
Holomorphic banach vector bundles on the maximal ideal. Dec 10, 2015 then by simply composing, a cocycle gij defining a rank n bundle, yields because f preserves compositions another cocycle fgij defining a rank m bundle. Induced bundles and vector bundles in the previous lecture, we have seen how principal g bundles correspond to equivalence classes of cocycles. Pdf vector bundles from generalized pairs of cocycles. Accordingly, the local triviality condition in the definition of topological vector bundles def. I have reformatted this with narrower margins for a better reading experience on devices like an ipad, but for a paper copy with more standard size margins try printing at 8590 per cent of full size. Connections on a given vector bundle e b make an a. Continuous vector bundles 405 coesficients in the traditional sense of the word. Continuous vector bundles over topological algebras, i i. The punchline of this worksheet is that the set of line bundles on a given variety x forms an group under the operation of. Whenever one studies moduli spaces of vector bundles on curves, one of the first things to be introduced is the slope of a vector bundle, i. Linear algebra operations on vector bundles motivation. A smooth real vector bundle of rank k over the base manifold m is a manifold e called the total space, together with a smooth surjection e m called the bundle projection, such that. A vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space x, where x is a topological space or a manifold or an algebraic variety.
Standard references for vector bundles and connections are ghv and kn, where the interested. In this case ed is a holomorphic banach vector bundle on d in the usual. In particular, the two gl1cocycles are contained in a. In smooth dynamics linear cocycles arise naturally from the derivative. I must admit, however, that the appreciation of this beauty came to physicists only in recent years. It is a real vector bundle of rank n dimm which in general is not trivial. One can recover the ber bundle from this principal bundle by a simple construction of. The space xis called the base of the vector bundle, and eis called the total space of the vector bundle. 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. M m is an automorphism fof a vector bundle e over m that covers f. Also, the vector spaces are usually required to be over the real or complex numbers, in which case the vector bundle is said to be a real or complex vector bundle respectively.
With vector bundles, we are interested in sections of the projection. Motivation let x,o be a cp premanifold with corners, 0. We would like to show you a description here but the site wont allow us. A vector bundle for which a trivialization exists is called trivializable. I guess what im confused about is where that condition comes from in cohomology. The cocycle definition leads to the same vector bundle e as discussed before. Two questions for vector bundles thread starter 1591238460. The euler class, the thom class and the chern classes are a few.
In mathematics, and particularly topology, a fiber bundle or, in british english, fibre bundle is a space that is locally a product space, but globally may have a different topological structure. Indeed, if e is a bundle represented by a holomorphic 1 cocycle gij. A morphism of vector bundles is a commutative diagram e 1. Characteristic classes assign to the each vector bundle a cohomology class of the base space. I understand that equation 42 in brylinski is the condition that two bundles are isomorphic. Consequently our results will have application to the study of those manifolds which admit such fiberings over tori. X such that for every x 2 x, there is an open neighborhood u of x and.
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