A vector bundle is determined its transition functions and give an open cover u. Vector bundles thus combine topology with linear algebra, and the study of vector bundles could be called linear algebraic topology. 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. I must admit, however, that the appreciation of this beauty came to physicists only in recent years. It is interesting to know, how far we can generalize the notion of a groupvalued cocycle keeping the property to determine a bundle. B, onto b, is called a vector bundle of rank kover bif the following holds. We would like to show you a description here but the site wont allow us. If the dimension of the vector space is mthen the bundle is often called an mplane bundle. In this paper we take f to be a transitive anosov di. The cocycle definition leads to the same vector bundle e as discussed before. 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.
These functions satisfy a cocycle condition, and we can use them to glue together e. Pdf vector bundles from generalized pairs of cocycles. Vector bundles in algebraic geometry enrique arrondo notes prepared for the first summer school on complex geometry villarrica, chile 79 december 2010 1. I guess what im confused about is where that condition comes from in cohomology. 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. S2 of rank 2 for each k2z or a complex vector bundle of rank 1, since g 01.
M m is an automorphism fof a vector bundle e over m that covers f. Linear algebra operations on vector bundles motivation. Induced bundles and vector bundles in the previous lecture, we have seen how principal gbundles correspond to equivalence classes of cocycles. Insection3,weshall be interested in the much more involved classi. In mathematics, a vector bundle is a topological construction that makes precise the idea of a. The cocycle construction of dual bundle is easy for definition but it does not lie in my head solidly. Holomorphic banach vector bundles on the maximal ideal space. 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.
The punchline of this worksheet is that the set of line bundles on a given variety x forms an group under the operation of. Connections on a given vector bundle e b make an a. The space xis called the base of the vector bundle, and eis called the total space of the vector bundle. Accordingly, the local triviality condition in the definition of topological vector bundles def. Finite group actions on moduli spaces of vector bundles florent schaffhauser abstract. Vector bundles and connections the exposition of vector. In particular, the two gl1cocycles are contained in a. Dual of a holomorphic vector bundle mathematics stack exchange. In particular, we show that for trivial bundles the oka principle is valid. Let v be a real or complex vector space, and let glv be the group of. Finite group actions on moduli spaces of vector bundles arxiv. A smooth real vector bundle v of rank k over a smooth manifold m is a.
Thus there exists an open cover of u fugso that mj u is trivial when restrict to u2u. The isomorphism class of a vector bundle constructed from a glk cocycle depends only on the equivalence class of the cocycle. A smooth real vector bundle of rank k over the base manifold m is a manifold e. Other kinds of space may be attached, thus entering the more general theory of fibre bundles. So the sheaf of groups in question will be that of germs of continuous gln, avalued. Continuous vector bundles 405 coesficients in the traditional sense of the word. It is a real vector bundle of rank n dimm which in general is not trivial. There are no triple overlaps, so the cocycle condition is vacuously satis. We we say that two vector bundles over m are isomorphic if there is a bundle map. 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. 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. 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. In order to help motivate them, we rst discuss vector bundles, and how they.
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. The tangent bundle tm is indeed a vector bundle, of rank dimm. Sections of a vector bundle reveal information about how trivial the bundle is. Rd, a linear cocycle can be represented by a matrixvalued function a. Introduction to finite and infinite dimensional lie superalgebras, 2016. A compact complex manifold m is called a hodge manifold if there exists a positive line bundle l over m. 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. The last condition is known as the cech cocycle condition more. 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. In this case ed is a holomorphic banach vector bundle on d in the usual. Then 1 there is an ainvariant, pointwise decomposition of v. The transition function satisfy the cocycle condition.
A bundle over a manifold is trivial if it is simply the cartesian product of. Nevertheless, since p is locally trivial, local sections do exist. Characteristic classes assign to the each vector bundle a cohomology class of the base space. Continuous vector bundles over topological algebras, i i. This seminar is an account of alan weinsteins recent paper the volume of a differentiable stack. 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. We will consider rst the simplest theory of vector bundles where the vector space is a onedimensional complex vector space line bundles. A section of a fiber bundle is a function whose output range is continuously dependent on the input. Induced bundles and vector bundles induced bundles. A vector bundle with fiber v is a triple e, b, p where e and b are smooth. Bring vector space back in picture cliff construct manifolds from given manifolds understand manifolds i. The only two vector bundles with base space a circle and onedimensional. An isomorphism is a morphism of line bundles that admits an inverse.
We find a generalization for pairs of cocycles and show how these generalized pairs of cocycles can still determine vector bundles. Motivation let x,o be a cp premanifold with corners, 0. A vector bundle for which a trivialization exists is called trivializable. The euler class, the thom class and the chern classes are a few.
Vector bundles are almost always required to be locally trivial, however, which means they are examples of fiber bundles. 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. Two questions for vector bundles thread starter 1591238460. I understand that equation 42 in brylinski is the condition that two bundles are isomorphic. 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. X such that for every x 2 x, there is an open neighborhood u of x and. Intuitively this is the object we get by gluing at each point p. Holomorphic banach vector bundles on the maximal ideal. Induced bundles and vector bundles in the previous lecture, we have seen how principal g bundles correspond to equivalence classes of cocycles. Conversely, any holomorphic matrix cocycle inscribed in an open covering of t determines a holomorphic vector bundle over t.
Standard references for vector bundles and connections are ghv and kn, where the interested. Almost synonymous terms used in various areas are topological bundle, locally trivial fibre bundle, fibre space, fibration, skew product etc. One can recover the ber bundle from this principal bundle by a simple construction of. Specifically, under mild technical assumptions a fiber bundle always has the homotopy lifting property or homotopy covering property see steenrod 1951, 11. The upshot of this is that ndimensional vector bundles are classi ed by nonabelian cohomology h1x.
This homotopy fiber may be thought of as the internal principal. With vector bundles, we are interested in sections of the projection. A morphism of vector bundles is a commutative diagram e 1. Consequently our results will have application to the study of those manifolds which admit such fiberings over tori. Vector bundles, connections and curvature 3 there is an obvious projection t. In smooth dynamics linear cocycles arise naturally from the derivative.
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