Christoph Wockel

University of Hamburg, Hamburg, Hamburg, Germany

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Publications (24)12.59 Total impact

  • Alexander Schmeding · Christoph Wockel
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    ABSTRACT: To a Lie groupoid over a compact base, the associated group of bisection is an (infinite-dimensional) Lie group. Moreover, under certain circumstances one can reconstruct the Lie groupoid from its Lie group of bisections. In the present article we consider functorial aspects of these construction principles. The first observation is that this procedure is functorial (for morphisms fixing the base). Moreover, it gives rise to an adjunction between the category of Lie groupoids over a fixed base and the category of Lie groups acting on the base. In the last section we then show how to promote this adjunction to almost an equivalence of categories.
    No preview · Article · Jun 2015
  • Alexander Schmeding · Christoph Wockel
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    ABSTRACT: This paper is about the relation of the geometry of Lie groupoids over a fixed compact manifold and the geometry of their (infinite-dimensional) bisection Lie groups. In the first part of the paper we investigate the relation of the bisections to a given Lie groupoid, where the second part is about the construction of Lie groupoids from candidates for their bisection Lie groups. The procedure of this second part becomes feasible due to some recent progress in the infinite-dimensional Frobenius theorem, which we heavily exploit. The main application to the prequantisation of (pre)symplectic manifolds comes from an integrability constraint of closed Lie subalgebras to closed Lie subgroups. We characterise this constraint in terms of a modified discreteness conditions on the periods of that manifold.
    No preview · Article · Jun 2015
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    Alexander Schmeding · Christoph Wockel
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    ABSTRACT: In this article we endow the group of bisections of a Lie groupoid with compact base with a natural locally convex Lie group structure. Moreover, we develop thoroughly the connection to the algebra of sections of the associated Lie algebroid and show for a large class of Lie groupoids that their groups of bisections are regular in the sense of Milnor.
    Preview · Article · Sep 2014 · Annals of Global Analysis and Geometry
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    Christoph Wockel
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    ABSTRACT: In this paper we analyse the topological group cohomology of finite-dimensional Lie groups. We introduce a technique for computing it (as abelian groups) for torus coefficients by the naturally associated long exact sequence. The upshot in there is that certain morphisms in this long exact coefficient sequence can be accessed (at least for semi-simple Lie groups) very conveniently by the Chern-Weil homomorphism of the naturally associated compact dual symmetric space. Since the latter is very well-known, this gives the possibility to compute the topological group cohomology of the classical simple Lie groups. In addition, we establish a relation to characteristic classes of flat bundles.
    Preview · Article · Jan 2014
  • Holomorphic Mappings · Christoph Wockel
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    ABSTRACT: In this paper we present another notion of a smooth manifold with corners and relate it to the commonly used concept in the literature. Afterwards we introduce complex manifolds with corners show that if M is a compact (respectively complex) manifold with corners and K is a smooth (respectively complex) Lie group, then C1(M, K) (respectively C1(M, K)) is a smooth (respectively complex) Lie group.
    No preview · Article · Jan 2013
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    Martin Fuchssteiner · Christoph Wockel
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    ABSTRACT: We show that for topological groups and loop contractible coefficients the cohomology groups of continuous group cochains and of group cochains that are continuous on some identity neighbourhood are isomorphic. Moreover, we show a similar statement for compactly generated groups and Lie groups holds and apply our results to different concepts of group cohomology for finite-dimensional Lie groups.
    Preview · Article · Jul 2012 · Topology and its Applications
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    Chenchang Zhu · Christoph Wockel
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    ABSTRACT: The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of \'etale Lie 2-groups. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of \pi_2 for each finite-dimensional Lie group. This fact was used by Cartan (in a slightly other guise) to construct the simply connected Lie group associated to each finite-dimensional Lie algebra. In infinite dimensions, there is an obstruction for a central extension of Lie algebras to integrate to a central extension of Lie groups. This obstruction comes from non-trivial \pi_2 for general Lie groups. We show that this obstruction may be overcome by integrating central extensions of Lie algebras not to Lie groups but to central extensions of \'etale Lie 2-groups. As an application, we obtain a generalization of Lie's Third Theorem to infinite-dimensional Lie algebras.
    Preview · Article · Apr 2012
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    Christoph Wockel
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    ABSTRACT: Lieʼs Third Theorem, asserting that each finite-dimensional Lie algebra is the Lie algebra of a Lie group, fails in infinite dimensions. The modern account on this phenomenon is the integration problem for central extensions of infinite-dimensional Lie algebras, which in turn is phrased in terms of an integration procedure for Lie algebra cocycles.This paper remedies the obstructions for integrating cocycles and central extensions from Lie algebras to Lie groups by generalising the integrating objects. Those objects obey the maximal coherence that one can expect. Moreover, we show that they are the universal ones for the integration problem.The main application of this result is that a Mackey-complete locally exponential Lie algebra (e.g., a Banach–Lie algebra) integrates to a Lie 2-group in the sense that there is a natural Lie functor from certain Lie 2-groups to Lie algebras, sending the integrating Lie 2-group to an isomorphic Lie algebra.
    Preview · Article · Nov 2011 · Advances in Mathematics
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    Friedrich Wagemann · Christoph Wockel
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    ABSTRACT: We propose a unified framework in which the different constructions of cohomology groups for topological and Lie groups can all be treated on equal footings. In particular, we show that the cohomology of "locally continuous" cochains (respectively "locally smooth" in the case of Lie groups) fits into this framework, which provides an easily accessible cocycle model for topological and Lie group cohomology. We illustrate the use of this unified framework and the relation between the different models in various applications. This includes the construction of cohomology classes characterizing the string group and a direct connection to Lie algebra cohomology.
    Preview · Article · Oct 2011 · Transactions of the American Mathematical Society
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    Karl-Hermann Neeb · Friedrich Wagemann · Christoph Wockel
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    ABSTRACT: If P \to X is a topological principal K-bundle and \hat K a central extension of K by Z, then there is a natural obstruction class \delta_1(P) in \check H^2(X,\uline Z) in sheaf cohomology whose vanishing is equivalent to the existence of a \hat K-bundle \hat P over X with P \cong \hat P/Z. In this paper we establish a link between homotopy theoretic data and the obstruction class \delta_1(P) which in many cases can be used to calculate this class in explicit terms. Writing \partial_d^P \: \pi_d(X) \to \pi_{d-1}(K) for the connecting maps in the long exact homotopy sequence, two of our main results can be formulated as follows. If Z is a quotient of a contractible group by the discrete group \Gamma, then the homomorphism \pi_3(X) \to \Gamma induced by \delta_1(P) \in \check H^2(X,\uline Z) \cong H^3_{\rm sing}(X,\Gamma) coincides with \partial_2^{\hat K} \circ \partial_3^P and if Z is discrete, then \delta_1(P) \in \check H^2(X,\uline Z) induces the homomorphism -\partial_1^{\hat K} \circ \partial_2^P \: \pi_2(X) \to Z. We also obtain some information on obstruction classes defining trivial homomorphisms on homotopy groups.
    Full-text · Article · Aug 2011 · Proceedings of the London Mathematical Society
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    Thomas Nikolaus · Christoph Sachse · Christoph Wockel
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    ABSTRACT: 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 establish some facts on the homotopy theory of Lie 2-groups. Moreover, we provide an explicit comparison of string structures for the two models and a uniqueness result for Lie 2-group models.
    Preview · Article · Apr 2011 · International Mathematics Research Notices
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    Bas Janssens · Christoph Wockel
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    ABSTRACT: We show that the canonical central extension of the group of sections of a Lie group bundle over a compact manifold, constructed in [NW09], is universal. In doing so, we prove universality of the corresponding central extension of Lie algebras in a slightly more general setting.
    Full-text · Article · Oct 2010 · Journal für die reine und angewandte Mathematik (Crelles Journal)
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    Christoph Wockel
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    ABSTRACT: In this paper we introduce principal 2-bundles and show how they are classified by non-abeliaň Cech cohomology. Moreover, we show that their gauge 2-groups can be described by 2-group-valued functors, much like in classical bundle theory. Using this, we show that, under some mild requirements, these gauge 2-groups possess a natural smooth structure. In the last section we provide some explicit examples.
    Preview · Article · Oct 2009 · Forum Mathematicum
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    Christoph Wockel
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    ABSTRACT: It is well-known that the central extensions of the loop group of a compact, simple and 1-connected Lie group are parametrised by their level $k \in Z$. This article concerns the question how much can be said for arbitrary $k \in R$ and we show that for each $k$ there exists a Lie groupoid which has the level $k$ central extension as its quotient if $k \in Z$. By considering categorified principal bundles we show, moreover, that the corresponding Lie groupoid has the expected bundle structure.
    Preview · Article · Oct 2009
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    Christoph Sachse · Christoph Wockel
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    ABSTRACT: Using the categorical description of supergeometry we give an explicit construction of the diffeomorphism supergroup of a compact finite-dimensional supermanifold. The construction provides the diffeomorphism supergroup with the structure of a Frechet supermanifold. In addition, we derive results about the structure of diffeomorphism supergroups.
    Preview · Article · Apr 2009 · Advances in Theoretical and Mathematical Physics
  • Christoph Wockel
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    ABSTRACT: Lie's Third Theorem, asserting that each finite-dimensional Lie algebra is the Lie algebra of a Lie group, fails in infinite dimensions. The modern account on this phenomenon is the integration problem for central extensions of infinite-dimensional Lie algebras, which in turn is phrased in terms of an integration procedure for Lie algebra cocycles. This paper remedies the obstructions for integrating cocycles and central extensions from Lie algebras to Lie groups by generalising the integrating objects. Those objects obey the maximal coherence that one can expect. Moreover, we show that they are the universal ones for the integration problem. The main application of this result is that a Mackey-complete locally exponential Lie algebra (e.g., a Banach-Lie algebra) integrates to a Lie 2-group in the sense that there is a natural Lie functor from certain Lie 2-groups to Lie algebras, sending the integrating Lie 2-group to an isomorphic Lie algebra.
    No preview · Article · Dec 2008
  • Christoph Wockel
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    ABSTRACT: In this paper we introduce gerbes as semi-strict principal 2-bundles and show how they are parametrised by non-abelian ÿ Cech cohomology. Moreover, we derive their gauge 2-groups (or gauge stacks) from first principles and show that they can be described by group-valued functors, much like in classical bundle theory. Moreover, we show that these 2- groups of group valued functors possess a natural smooth structure (under some mild requirements on the structure group). In the last section we provide some explicit examples. MSC: 55R65, 22E65, 81T13
    No preview · Article · Jan 2008
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    C. Wockel
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    ABSTRACT: This paper is on the connecting homomorphism in the long exact homotopy sequence of the evaluation fibration evp0 :C(P, K)K →K, whereC(P, K)K is the gauge group of a continuous principalK-bundle. We show that in the case of a bundle over a sphere or a orientable surface the connecting homomorphism is given in terms of the Samelson product. As applications we get an explicit formula for π2(C(P k ,K)K ), whereP k denotes the principal S3-bundle over S4 of Chern numberk and derive explicit formulae for the rational homotopy groups πn (C(P,K)K )⊗ℚ.
    Preview · Article · Dec 2007 · Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
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    Karl-Hermann Neeb · Christoph Wockel
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    ABSTRACT: If q : P -> M is a principal K-bundle over the compact manifold M, then any invariant symmetric V-valued bilinear form on the Lie algebra k of K defines a Lie algebra extension of the gauge algebra by a space of bundle-valued 1-forms modulo exact forms. In the present paper we analyze the integrability of this extension to a Lie group extension for non-connected, possibly infinite-dimensional Lie groups K. If K has finitely many connected components we give a complete characterization of the integrable extensions. Our results on gauge groups are obtained by specialization of more general results on extensions of Lie groups of smooth sections of Lie group bundles. In this more general context we provide sufficient conditions for integrability in terms of data related only to the group K. Comment: 54 pages, revised version, to appear in Ann. Glob. Anal. Geom
    Full-text · Article · Nov 2007 · Annals of Global Analysis and Geometry
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    Christoph Wockel
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    ABSTRACT: In this paper we describe how one can obtain Lie group structures on the group of (vertical) bundle automorphisms for a locally convex principal bundle P over the compact manifold M. This is done by first considering Lie group structures on the group of vertical bundle automorphisms Gau(P). Then the full automorphism group Aut(P) is considered as an extension of the open subgroup DiffP(M) of diffeomorphisms of M preserving the equivalence class of P under pull-backs, by the gauge group Gau(P). We derive explicit conditions for the extensions of these Lie group structures, show the smoothness of some natural actions and relate our results to affine Kac–Moody algebras and groups.
    Preview · Article · Oct 2007 · Journal of Functional Analysis