By the cocycle condition and the choice of c we see that x. If we suppress and simply denote the action by, the condition is. A kvector bundle isomorphic to a product bundle is called a trivial bundle. More generally, a glkcocycle on mis an open cover fv g 2a of malong with maps. To build a vector bundle, choose an open cover ui and form the pieces ui rk then glue these together on double overlaps uij via functions gij. Our transition maps probably dont need to take values in the whole di. 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. Induced bundles and vector bundles induced bundles. Induced bundles and vector bundles since s ix ijx s jx, we nd that 0 ijxy x y. Chapter 7 vector bundles louisiana state university.
A 2cocycle for the action is a function satisfying. Equivalently, it is a line bundle on m 1 satisfying the cocycle condition l 01 l 12 l 20 1 where 1 is the trivial bundle. The main tool of the investigation in finsler geometry is the finsler connection. To build a vector bundle, choose an open cover ui and form the pieces ui. Note that a function without any conditions is sometimes termed a 2cochain for the group action. A bundle over a manifold is trivial if it is simply the cartesian product of. M 1 with a multiplication l 01 l 12 l 02 satisfying a certain compatibility condition on m 3 here by l ij we mean the pullback of l to m 2 via the appropriate face map. A vector bundle is determined its transition functions and give an open cover u. If this torsion twisting is nontrivial there is no, locally.
Let e and b differentiable manifolds and e b a smooth map. This statement is also known as the ehresmann theorem, see ehresmann, c. So we obtain a glkcocycle on m from any vector bundle of rank k on m namely, the structure. A vector bundle of rank k over a manifold m is a manifold e and a. We would like to show you a description here but the site wont allow us. Definition as part of the general definition of cocycle. 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. If we suppress and use for the action, we can rewrite this as. It is a real vector bundle of rank n dimm which in general is not trivial. Then f p cand hence, again by the cocycle condition f p c whenever p2u. A smooth real vector bundle v of rank k over a smooth manifold m is a. One can recover the ber bundle from this principal bundle by a simple construction of. The cocycle definition leads to the same vector bundle e as discussed before.
The trivial line bundle corresponds to a cocycle fg uv satisfying g uv g ug 1 v where fg ugis a set of nonvanishing smooth functions on u2u. These are lecture notes of a short course on the moduli stack of vector bundles on an algebraic curve. A holomorphic gerbe is analogously defined by functions h. Robertgeroch, 1969 in this lecture we will discuss the notion of a spin structure on a. Lectures on the moduli stack of vector bundles on a curve. M, 1pep has the structure of kdimensional vector space, each p.
This is called the cocycle condition, which we will examine in a moment. A vector bundle for which a trivialization exists is called trivializable. The euler class, the thom class and the chern classes are a few. C where c is the sheaf of germs of nonvanishing smooth functions. Let be a group acting on the left on an abelian group via a homomorphism of groups where is the automorphism group of explicit definition. Pdf vector bundles from generalized pairs of cocycles.
For instance if you take the big zariski site, then an object with descent data is just a scheme. If the dimension of the vector space is mthen the bundle is often called an mplane bundle. Spinmanifolds thus, the existence of a spinor structure appears, on physical grounds, to be a reasonable condition to impose on any cosmological model in general relativity. Vector bundles a vector bundle is a special case of a.
Cohomologous cocycles give isomorphic bundles and conversely. In other words, the new ber bundle p gfis a ber bundle with group g, and the same cocycle as p. It is the same idea as the cocycle equation for gluing together a scheme from affine open subschemes. A vector bundle over xis a ber bundle with ber v and structure group glv. Characteristic classes assign to the each vector bundle a cohomology class of the base space. From this, it is clear that, in condition ii, the rst coordinate of ijx.
The fubini principle implies that if one does this a second time with the roles of the two operators interchanged, then the resulting. Pdf the cocycle condition for multipullbacks of algebras. 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. A 1cocycle for a group action is a special case of a cocycle for a group action in the case. Let v be a real or complex vector space, and let glv be the group of.
More generally, a glk cocycle on mis an open cover fv g 2a of malong with maps. In this every object is fibrant, and the cofibrant objects. The importance of this is that the transition functions determine the fiber bundle if one assumes the cech cocycle condition. In this lecture we introduce the notion of a principal.
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. The connection is a unique one on the hermitian bundle ve. This, in turn, is the notion of cocycle corresponding to the hom complex from the bar resolution of to as modules. Almost synonymous terms used in various areas are topological bundle, locally trivial fibre bundle, fibre space, fibration, skew product etc. So we obtain a glkcocycle on m from any vector bundle of rank kon m namely, the structure cocycle. Accordingly, the local triviality condition in the definition of topological vector bundles def.
In fact, if the bundle were trivial, the total space would be disconnected, being two disjoint copies of the circle. Then f p cand hence, again by the cocycle condition f p. Therefore this section is devoted to the notion of vector bundles. Tn also written tf when in a categorical mood is the collection of. Let v be a real or complex vector space, and let glv be the group of linear isomorphisms of v. Finally, if p2v, let u 2u be an open set containing pu exists because u is a cover.
However building a paper model of the mobius band one quickly sees that its boundary is connected. Let 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. The last condition is known as the cech cocycle condition more. The equivariant cocycle condition then ensures that the elements. As long as g ij satisfy g ij g 1 ji as well as the cocycle condition, the resulting space has a. Induced bundles and vector bundles in the previous lecture, we have seen how principal gbundles correspond to equivalence classes of cocycles.
The nonexistense of the fine moduli scheme of vector bundles. The derivation of the equivariant cocycle condition uses the fact that the left and right actions commute. Let v 2tmm be represented as the velocity of a curve t 7. The notion of vector bundles is going to be very important for other chapters of this book. B, onto b, is called a vector bundle of rank kover bif the following holds. For a vector bundle, we only need the general linear group glnr. Then one can pair the longitudinal cocycle for one of these operators with the transverse cocycle for the other to get a new cocycle. Roughly speaking, a vector bundle e can be thought of as a manifold m with a vector space e p attached to each point p. It is also sometimes convenient to consider inniterank vector bundles. We can understand this bundle in terms of the local data as follows. My question is about why the condition below is called the cocycle condition. A 1 cocycle for a group action is a special case of a cocycle for a group action in the case. So we obtain a glk cocycle on m from any vector bundle of rank kon m namely, the structure cocycle.
In this paper we take f to be a transitive anosov di. The trivial line bundle corresponds to a cocycle fg uv satisfying g uv g ug 1 v. In this denition, n is called the rank of the vector bundle. Surely it is named after some interpretation of it in cohomology. A vector bundle with fiber v is a triple e, b, p where e and b are smooth manifolds. Lectures on the moduli stack of vector bundles on a curve jochen heinloth preliminary version, january 29, 2009 abstract. To build a vector bundle, choose an open cover fu igand form the pieces fu i rkgthen glue these together on double overlaps fu ijgvia functions g ij. Pdf vector bundles and their applications researchgate. The longitudinal cocycle and the index of toeplitz operators. A holomorphic line bundle is defined by transition functions relative to open sets u.
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