Publications (8)0 Total impact
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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.
10/2011;
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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.
10/2011;
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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.
08/2011;
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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.
04/2011;
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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.
10/2010;
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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.
04/2009;
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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
11/2007;
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ABSTRACT: Let K be a 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.
04/2006;
