Equivalences of smooth and continuous principal bundles with infinite-dimensional structure group

@article{Muller2006EquivalencesOS,
  title={Equivalences of smooth and continuous principal bundles with infinite-dimensional structure group},
  author={Christophe Muller and Christoph Wockel},
  journal={Advances in Geometry},
  year={2006},
  volume={9}
}
Let K be a Lie group, modeled on a locally convex space, and M a finite-dimensional paracompact manifold with corners. We show that each continuous principal K-bundle over M is continuously equivalent to a smooth one and that two smooth principal K-bundles over M which are continuously equivalent are also smoothly equivalent. In the concluding section, we relate our results to neighboring topics. 
View PDF on arXiv
33 Citations
Highly Influential Citations
5
Background Citations
8
Methods Citations
4
Results Citations
1

33 Citations

Connections on central extensions, lifting gerbes, and finite-dimensional obstruction vanishing

Given a central extension of Lie groups, we study the classification problem of lifting the structure group together with a given connection. For reductive structure groups we introduce a new…

Smooth Models for Certain Fibered Partially Hyperbolic Systems

. We prove that under restrictions on the fiber, any fibered partially hyperbolic system over a nilmanifold is leaf conjugate to a smooth model that is isometric on the fibers and descends to a…

Non-trivial smooth families of K3 surfaces

. Let X be a complex K 3 surface, Diff( X ) the group of diffeomorphisms of X and Diff 0 ( X ) the identity component. We prove that the fundamental group of Diff 0 ( X ) contains a free abelian group of…

Lie group structures on groups of smooth and holomorphic maps on non-compact manifolds

We study Lie group structures on groups of the form C∞(M, K), where M is a non-compact smooth manifold and K is a, possibly infinite-dimensional, Lie group. First we prove that there is at most one…

O A ] 1 0 A ug 2 00 7 KK-THEORETIC DUALITY FOR PROPER TWISTED ACTIONS

Let A be a smooth continuous trace algebra, with a Riemannian manifold spectrumX, equipped with a smooth action by a discrete group G such that G acts onX properly and isometrically. Then A⋊G is…

FOUR EQUIVALENT VERSIONS OF NONABELIAN GERBES

We recall and partially expand four versions of smooth, non-abelian gerbes: Cech cocycles, classifying maps, bundle gerbes, and principal 2-bundles. We prove that all these four versions are…

A Smooth Model for the String Group

We construct a model for the string group as an infinite-dimensional Lie group. In a second step we extend this model by a contractible Lie group to a Lie 2-group model. To this end we need to…

S ep 2 00 7 Lie groups of bundle automorphisms and their extensions

We describe natural abelian extensions of the Lie algebra aut(P ) of infinitesimal automorphisms of a principal bundle over a compact manifold M and discuss their integrability to corresponding Lie…

Constraints on families of smooth 4–manifolds from Bauer–Furuta invariants

We obtain constraints on the topology of families of smooth $4$-manifolds arising from a finite dimensional approximation of the families Seiberg-Witten monopole map. Amongst other results these…

Lie Groups of Bundle Automorphisms and Their Extensions

We describe natural abelian extensions of the Lie algebra \(\mathfrak{aut}(P)\) of infinitesimal automorphisms of a principal bundle over a compact manifold M and discuss their integrability to…

References

SHOWING 1-10 OF 41 REFERENCES

Fundamentals of direct limit Lie theory

We show that every countable direct system of finite-dimensional real or complex Lie groups has a direct limit in the category of Lie groups modelled on locally convex spaces. This enables us to push…

Topology of Fibre Bundles

Fibre bundles, an integral part of differential geometry, are also important to physics. This text, a succint introduction to fibre bundles, includes such topics as differentiable manifolds and…

The Convenient Setting of Global Analysis

Introduction Calculus of smooth mappings Calculus of holomorphic and real analytic mappings Partitions of unity Smoothly realcompact spaces Extensions and liftings of mappings Infinite dimensional…

A GENERALIZATION OF STEENROD'S APPROXIMATION THEOREM

In this paper we aim for a generalization of the Steenrod Approximation Theorem from [16, Section 6.7], concerning a smoothing procedure for sections in smooth locally trivial bundles. The…

Infinite-dimensional Lie groups without completeness restrictions

We describe a setting of infinite-dimensional smooth (resp., analytic) Lie groups modelled on arbitrary, not necessarily sequentially complete, locally convex spaces, generalizing the framework of…

SMOOTH AND CONTINUOUS HOMOTOPIES INTO CONVENIENT MANIFOLDS AGREE

Continuous and smooth homotopies agree from smooth finite dimensional manifolds into infinite dimensional ones which are modeled on convenient vector spaces. Since convex charts do not exist we use…

Twisted K-Theory and K-Theory of Bundle Gerbes

Abstract: In this note we introduce the notion of bundle gerbe K-theory and investigate the relation to twisted K-theory. We provide some examples. Possible applications of bundle gerbe K-theory to…

On a generalised Connes–Hochschild–Kostant–Rosenberg theorem

Principal Bundles and the Dixmier Douady Class

Abstract:A systematic consideration of the problem of the reduction and “lifting” of the structure group of a principal bundle is made and a variety of techniques in each case are explored and…

Introduction to Smooth Manifolds

Preface.- 1 Smooth Manifolds.- 2 Smooth Maps.- 3 Tangent Vectors.- 4 Submersions, Immersions, and Embeddings.- 5 Submanifolds.- 6 Sard's Theorem.- 7 Lie Groups.- 8 Vector Fields.- 9 Integral Curves…