Introduction to Foliations and Lie Groupoids - download pdf or read online

By I. Moerdijk, J. Mrcun

ISBN-10: 0521831970

ISBN-13: 9780521831970

In accordance with a graduate path taught at Utrecht college, this publication presents a quick creation to the speculation of Foliations and Lie Groupoids to scholars who've already taken a primary direction in differential geometry. Ieke Moerdijk and Janez Mrcun comprise special references to let scholars to discover the considered necessary history fabric within the study literature. The textual content positive factors many routines and labored examples.

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Now observe that, by compactness, σ hits each leaf finitely often. Choose a leaf L which is hit by σ , and let z0 and z1 be two consecutive 48 Holonomy and stability Fig. 3. Modification of γ points on S 1 with σ (z0 ), σ (z1 ) ∈ L. Choose a simple smooth path β in L from y = σ (z1 ) to x = σ (z0 ). 4). Note that by the same argument as before, all the leaves of F hit by σ are diffeomorphic to L0 . Fig. 4. Modification of σ For any z ∈ S 1 let Lz be the leaf with σ(z) ∈ Lz . The leaf Lz is compact with trivial holonomy, so the foliation of a small saturated open neighbourhood V of Lz looks like the product Lz × (− , ).

12 (i) we can assume that W is in fact a G-stable subset of U and λ the inclusion, and we can also assume that x ∈ W ⊂ Z. 11. This proves that f is a local diffeomorphism. If Z = U , the facts that φ is proper onto its image and V Hausdorff imply that f is a proper local diffeomorphism onto its image, and hence a covering projection. 11. For the second part of the statement observe that the restriction of f to a G-stable open set is an embedding between orbifold charts. An orbifold atlas of dimension n of a topological space Q is a collection of pairwise compatible orbifold charts U = {(Ui , Gi , φi )}i∈I of dimension n on Q such that i∈I φi (Ui ) = Q.

It is a quotient group of the fundamental group of the leaf through x. This group contains a lot of information about the structure of the foliation around the leaf through x, especially if that leaf is compact. For example, if this group is finite then all the nearby leaves must also be compact, and the foliation locally looks like one which is obtained by the flat bundle construction from the previous chapter. 3 below. 5, we will present a ‘global’ stability theorem, which applies to foliations of codimension 1, and states that under certain conditions, the holonomy group has to be trivial and the foliation has to be simple (in the technical sense of being given by the fibres of a submersion).

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Introduction to Foliations and Lie Groupoids by I. Moerdijk, J. Mrcun

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