Publications (21)11.52 Total impact

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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. 
Article: Topological group cohomology of Lie groups and ChernWeil theory for compact symmetric spaces
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ABSTRACT: In this paper we analyse the topological group cohomology of finitedimensional 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 semisimple Lie groups) very conveniently by the ChernWeil homomorphism of the naturally associated compact dual symmetric space. Since the latter is very wellknown, 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. 
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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 finitedimensional Lie groups.Topology and its Applications 07/2012; 159(1011):2627–2634. DOI:10.1016/j.topol.2012.04.006 · 0.59 Impact Factor 
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ABSTRACT: The purpose of this paper is to show how central extensions of (possibly infinitedimensional) Lie algebras integrate to central extensions of \'etale Lie 2groups. 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 finitedimensional Lie group. This fact was used by Cartan (in a slightly other guise) to construct the simply connected Lie group associated to each finitedimensional 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 nontrivial \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 2groups. As an application, we obtain a generalization of Lie's Third Theorem to infinitedimensional Lie algebras. 
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ABSTRACT: Lieʼs Third Theorem, asserting that each finitedimensional 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 infinitedimensional 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 Mackeycomplete locally exponential Lie algebra (e.g., a Banach–Lie algebra) integrates to a Lie 2group in the sense that there is a natural Lie functor from certain Lie 2groups to Lie algebras, sending the integrating Lie 2group to an isomorphic Lie algebra.Advances in Mathematics 11/2011; 228(4):22182257. DOI:10.1016/j.aim.2011.07.003 · 1.35 Impact Factor 
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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.Transactions of the American Mathematical Society 10/2011; DOI:10.1090/S000299472014061072 · 1.10 Impact Factor 
Article: Making Lifting Obstructions Explicit
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ABSTRACT: If P \to X is a topological principal Kbundle 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 Kbundle \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_{d1}(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.Proceedings of the London Mathematical Society 08/2011; 106. DOI:10.1112/plms/pds047 · 1.12 Impact Factor 
Article: A Smooth Model for the String Group
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ABSTRACT: We construct a model for the string group as an infinitedimensional Lie group. In a second step we extend this model by a contractible Lie group to a Lie 2group model. To this end we need to establish some facts on the homotopy theory of Lie 2groups. Moreover, we provide an explicit comparison of string structures for the two models and a uniqueness result for Lie 2group models.International Mathematics Research Notices 04/2011; DOI:10.1093/imrn/rns154 · 1.07 Impact Factor 
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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.Journal für die reine und angewandte Mathematik (Crelles Journal) 10/2010; 682. DOI:10.1515/crelle20120021 · 1.30 Impact Factor 
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ABSTRACT: In this paper we introduce principal 2bundles and show how they are classified by nonabeliaň Cech cohomology. Moreover, we show that their gauge 2groups can be described by 2groupvalued functors, much like in classical bundle theory. Using this, we show that, under some mild requirements, these gauge 2groups possess a natural smooth structure. In the last section we provide some explicit examples.Forum Mathematicum 10/2009; 23(3). DOI:10.1515/FORM.2011.020 · 0.73 Impact Factor 
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ABSTRACT: It is wellknown that the central extensions of the loop group of a compact, simple and 1connected 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. 
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ABSTRACT: Using the categorical description of supergeometry we give an explicit construction of the diffeomorphism supergroup of a compact finitedimensional 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.Advances in Theoretical and Mathematical Physics 04/2009; DOI:10.4310/ATMP.2011.v15.n2.a2 · 1.78 Impact Factor 
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ABSTRACT: Lie's Third Theorem, asserting that each finitedimensional 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 infinitedimensional 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 Mackeycomplete locally exponential Lie algebra (e.g., a BanachLie algebra) integrates to a Lie 2group in the sense that there is a natural Lie functor from certain Lie 2groups to Lie algebras, sending the integrating Lie 2group to an isomorphic Lie algebra. 
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ABSTRACT: In this paper we introduce gerbes as semistrict principal 2bundles and show how they are parametrised by nonabelian ÿ Cech cohomology. Moreover, we derive their gauge 2groups (or gauge stacks) from first principles and show that they can be described by groupvalued 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 
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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 principalKbundle. 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 S3bundle over S4 of Chern numberk and derive explicit formulae for the rational homotopy groups π n (C(P,K) K )⊗ℚ.Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 12/2007; 77(1):219228. DOI:10.1007/BF03173500 · 0.22 Impact Factor 
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ABSTRACT: If q : P > M is a principal Kbundle over the compact manifold M, then any invariant symmetric Vvalued bilinear form on the Lie algebra k of K defines a Lie algebra extension of the gauge algebra by a space of bundlevalued 1forms modulo exact forms. In the present paper we analyze the integrability of this extension to a Lie group extension for nonconnected, possibly infinitedimensional 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. GeomAnnals of Global Analysis and Geometry 11/2007; DOI:10.1007/s1045500991686 · 0.79 Impact Factor 
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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 pullbacks, 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.Journal of Functional Analysis 10/2007; DOI:10.1016/j.jfa.2007.05.016 · 1.15 Impact Factor 
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ABSTRACT: In this paper we aim for a generalisation of the Steenrod Approximation Theorem from, concerning a smoothing procedure for sections in smooth locally trivial bundles. The generalisation is that we consider locally trivial smooth bundles with a possibly infinitedimensional typical fibre. The main result states that a continuous section in a smooth locally trivial bundles can always be smoothed out in a very controlled way (in terms of the graph topology on spaces of continuous functions), preserving the section on regions where it is already smooth.10/2006; 
Article: Equivalences of Smooth and Continuous Principal Bundles with InfiniteDimensional Structure Group
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ABSTRACT: Let K be a a Lie group, modeled on a locally convex space, and M a finitedimensional paracompact manifold with corners. We show that each continuous principal Kbundle over M is continuously equivalent to a smooth one and that two smooth principal Kbundles over M which are continuously equivalent are also smoothly equivalent. In the concluding section, we relate our results to neighboring topics.Advances in Geometry 04/2006; DOI:10.1515/ADVGEOM.2009.032 · 0.31 Impact Factor 
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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 and show that if $M$ is a compact (respectively complex) manifold with corners and $K$ is a smooth (respectively complex) Lie group, then $C^{\infty}(M,K)$ (respectively $C^{\infty}_{\C}(M,K)$) is a smooth (respectively complex) Lie group. Comment: 10 pages
Publication Stats
131  Citations  
11.52  Total Impact Points  
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Institutions

2010–2012

University of Hamburg
Hamburg, Hamburg, Germany


2007

GeorgAugustUniversität Göttingen
 Institute of Mathematics
Göttingen, Lower Saxony, Germany
