32 research outputs found

    Orbifold-like and proper g\mathfrak g-manifolds

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    In [4] and [5], we generalized the concept of completion of an infinitesimal group action ζ:g→X(M)\zeta : {\mathfrak g} \to \mathfrak X (M) to an actual group action on a (non-compact) manifold MM, originally introduced by R. Palais [9], and showed by examples that this completion may have quite pathological properties (much like the leaf space of a foliation). In the present paper, we introduce and investigate a tamer class of g\mathfrak g-manifolds, called orbifold--like, for which the completion has an orbifold structure. This class of g\mathfrak g-manifolds is reasonably well-behaved with respect to its local topological and smooth structure to allow for many geometric constructions to make sense. In particular, we investigate proper g\mathfrak g-actions and generalize many of the usual properties of proper group actions to this more general setting.Comment: 18 page

    Gerbes, simplicial forms and invariants for families of foliated bundles

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    The notion of a gerbe with connection is conveniently reformulated in terms of the simplicial deRham complex. In particular the usual Chern-Weil and Chern-Simons theory is well adapted to this framework and rather easily gives rise to `characteristic gerbes' associated to families of bundles and connections. In turn this gives invariants for families of foliated bundles. A special case is the Quillen line bundle associated to families of flat SU(2)-bundlesComment: 28 page

    Index theory for basic Dirac operators on Riemannian foliations

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    In this paper we prove a formula for the analytic index of a basic Dirac-type operator on a Riemannian foliation, solving a problem that has been open for many years. We also consider more general indices given by twisting the basic Dirac operator by a representation of the orthogonal group. The formula is a sum of integrals over blowups of the strata of the foliation and also involves eta invariants of associated elliptic operators. As a special case, a Gauss-Bonnet formula for the basic Euler characteristic is obtained using two independent proofs

    The eta invariant and equivariant index of transversally elliptic operators

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    We prove a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a G-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications, we obtain an index formula for basic Dirac operators on Riemannian foliations, a problem that was open for many years
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