Ulrich Bunke

A K-theoretic relative index theorem and Callias-type Dirac operators

We prove a relative index theorem for Dirac operators acting on sections of an A ? C-Cliiord bundle, where A is some C-algebra. We develop the index theory of complex and real Callias-type Dirac operators.

ON THE TOPOLOGY OF T-DUALITY

We study a topological version of the T-duality relation between pairs consisting of a principal U(1)-bundle equipped with a degree-three integral cohomology class. We describe the homotopy type of a classifying space for such pairs and show that it admits a selfmap which implements a T-duality transformation. We give a simple derivation of a T-duality isomorphism for certain twisted cohomology theories. We conclude with some explicit computations of twisted K-theory groups and discuss an example of iterated T-duality for higher-dimensional torus bundles.

Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group

We prove Pattersons conjecture about the singularities of the Selberg zeta function associated to a convex-cocompact, torsion free group acting on a hyperbolic space.

Index theory, eta forms, and Deligne cohomology

The Chern classes of a K-theory class which is represented by a vector bundle with connection admit refinements to Cheeger-Simons classes in Deligne cohomology. In the present paper we consider similar refinements in the case where the classes in K-theory are represented by geometric families of Dirac operators. In low dimensions these refinements correspond to the exponentiated eta-invariant, the determinant line bundle with Quillen metric and Bismut-Freed connection, and Lotts index gerbe with connection and curving. We give a unified treatement of these cases as well as their higher generalizations. Our main technical tool is a variant of local index theory for Dirac operators of families of manifolds with corners.

Relative index theory

Abstract We develop the framework for the heat equation method in the relative index theory. We study pairs of Dirac operators on complete manifolds and give an analytic interpretation of the difference of the integrals over their local index densities. This yields a generalization of the relative index theorem of Gromov and Lawson. Under some assumptions on the curvature we show, that supersymmetric scattering theories of Borisov, Muller, and Schrader arise naturally. The scattering index is related to the relative topological index.

Upper Bounds of small Eigenvalues of the Dirac Operator and Isometric immersions

We show that a topologically determined number of eigenvalues of the Dirac operatorD of a closed Riemannian spin manifoldM of even dimensionn can be bounded by the data of an isometric immersion ofM into the Euclidian spaceRN. From this we obtain similar bounds of the eigenvalues ofD in terms of the scalar curvature ofM ifM admits a minimal immersion intoSN or,ifM is complex, a holomorphic isometric immersion intoPCN.

Landweber exact formal group laws and smooth cohomology theories

The main aim of this paper is the construction of a smooth (sometimes called differential) extension \hat{MU} of the cohomology theory complex cobordism MU, using cycles for \hat{MU}(M) which are essentially proper maps W\to M with a fixed U(n)-structure and U(n)-connection on the (stable) normal bundle of W\to M. Crucial is that this model allows the construction of a product structure and of pushdown maps for this smooth extension of MU, which have all the expected properties. Moreover, we show, using the Landweber exact functor principle, that \hat{R}(M):=\hat{MU}(M)\otimes_{MU^*}R defines a multiplicative smooth extension of R(M):=MU(M)\otimes_{MU^*}R whenever R is a Landweber exact MU*-module. An example for this construction is a new way to define a multiplicative smooth K-theory.

A geometric description of differential cohomology@@@Une description géométrique de la cohomologie différentielle

In this paper we give a geometric cobordism description of smooth integral cohomology. The main motivation to consider this model (for other models see [4], [6], [5]) is that it allows for simple descriptions of both the cup product and the integration, so that it is easy to verify the compatibilty of these structures. We proceed in a similar way in the case of smooth cobordism as constructed in [3]. There the starting point was Quillen’s cobordism description of singular cobordism groups for a smooth manifold X. Here we use instead the similar description of ordinary cohomology from [9]. This cohomology theory is denoted by SHk(X). In this description smooth manifolds in Quillens’ description are replaced by so-called stratifolds, which are certain stratified spaces. The cohomology theory SHk(X) is naturally isomorphic to ordinary cohomology Hk(X), thus we obtain a cobordism type definition of the smooth extension of ordinary integral cohomology.

Duality for topological abelian group stacks and T-duality

We investigate when Isomorphism Conjectures, such as the ones due to Baum-Connes, Bost and Farrell-Jones, are stable under colimits of groups over directed sets (with not necessarily injective structure maps). We show in particular that both the K-theoretic Farrell-Jones Conjecture and the Bost Conjecture with coefficients hold for those groups for which Higson, Lafforgue and Skandalis have disproved the Baum-Connes Conjecture with coefficients.We present a

Sheaf theory for stacks in manifolds and twisted cohomology for S 1 -gerbes

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