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Ulrich Bunke

Publications and source records attributed to Ulrich Bunke.

At least 55 records · Page 3Linked to original sources
arXiv2018

Foliated manifolds, algebraic K-theory, and a secondary invariant

We introduce a $\mathbb{C}/\mathbb{Z}$-valued invariant of a foliated manifold with a stable framing and with a partially flat vector bundle. This invariant can be expressed in terms of integration in differential $K$-theory, or alternatively, in terms of $η$-invariants of Dirac operators and local correction terms. Initially, the construction of the element in $\mathbb{C}/\mathbb{Z}$ involves additional choices. But if the codimension of the foliation is sufficiently small, then this element is independent of these choices and therefore an invariant of the data listed above. We show that the invariant comprises various classical invariants like Adams' $e$-invariant, the $ρ$-invariant of twisted Dirac operators, or the Godbillon-Vey invariant from foliation theory. Using methods from differential cohomology theory we construct a regulator map from the algebraic $K$-theory of smooth functions on a manifold to its connective $K$-theory with $\mathbb{C}/\mathbb{Z}$ coefficients. Our main result is a formula for the invariant in terms of this regulator and integration in algebraic and topological $K$-theory.

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arXiv2018

Coarse homotopy theory and boundary value problems

We provide an interpretation of the APS index theorem of Piazza-Schick and Zeidler in terms of coarse homotopy theory. On the one hand we propose a motivic version of the boundary value problem, the index theorem, and the associated secondary invariants. On the other hand, we discuss in detail how the abstract version is related with classical case for Dirac operators.

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arXiv2018

The Beilinson regulator is a map of ring spectra

We prove that the Beilinson regulator, which is a map from $K$-theory to absolute Hodge cohomology of a smooth variety, admits a refinement to a map of $E_\infty$-ring spectra in the sense of algebraic topology. To this end we exhibit absolute Hodge cohomology as the cohomology of a commutative differential graded algebra over $\mathbb{R}$. The associated spectrum to this CDGA is the target of the refinement of the regulator and the usual $K$-theory spectrum is the source. To prove this result we compute the space of maps from the motivic $K$-theory spectrum to the motivic spectrum that represents absolute Hodge cohomology using the motivic Snaith theorem. We identify those maps which admit an $E_\infty$-refinement and prove a uniqueness result for these refinements.

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arXiv2017

A universal coarse K-theory

In this paper, we construct an equivariant coarse homology theory with values in the category of non-commutative motives of Blumberg, Gepner and Tabuada, with coefficients in any small additive category. Equivariant coarse K-theory is obtained from the latter by passing to global sections. The present construction extends joint work of the first named author with Engel, Kasprowski and Winges by promoting codomain of the equivariant coarse K-homology functor to non-commutative motives.

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arXiv2016

Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory

We develop differential algebraic K-theory for rings of integers in number fields and we construct a cycle map from geometrized bundles of modules over such a ring to the differential algebraic K-theory. We also treat some of the foundational aspects of differential cohomology, including differential function spectra and the differential Becker-Gottlieb transfer. We then state a transfer index conjecture about the equality of the Becker-Gottlieb transfer and the analytic transfer defined by Lott. In support of this conjecture, we derive some non-trivial consequences which are provable by independent means.

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arXiv2015

Multiplicative differential algebraic K-theory and applications

We construct a version of Beilinson's regulator as a map of sheaves of commutative ring spectra and use it to define a multiplicative variant of differential algebraic K-theory. We use this theory to give an interpretation of Bloch's construction of K_3-classes and the relation with dilogarithms. Furthermore, we provide a relation to Arakelov theory via the arithmetic degree of metrized line bundles, and we give a proof of the formality of the algebraic K-theory of number rings.

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arXiv2015

Regulators and cycle maps in higher-dimensional differential algebraic K-theory

We develop differential algebraic K-theory of regular arithmetic schemes. Our approach is based on a new construction of a functorial, spectrum level Beilinson regulator using differential forms. We construct a cycle map which represents differential algebraic K-theory classes by geometric vector bundles. As an application we derive Lott's relation between short exact sequences of geometric bundles with a higher analytic torsion form.

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arXiv2015

T-Duality via Gerby Geometry and Reductions

We consider topological T-duality of torus bundles equipped with S^{1}-gerbes. We show how a geometry on the gerbe determines a reduction of its band to the subsheaf of S^{1}-valued functions which are constant along the torus fibres. We observe that such a reduction is exactly the additional datum needed for the construction of a T-dual pair. We illustrate the theory by working out the example of the canonical lifting gerbe on a compact Lie group which is a torus bundles over the associated flag manifold. It was a recent observation of Daenzer and van Erp (arXiv1211.0763) that for certain compact Lie groups and a particular choice of the gerbe, the T-dual torus bundle is given by the Langlands dual group.

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arXiv2014

A regulator for smooth manifolds and an index theorem

For a smooth manifold X of dimension <d we construct a homomorphism from the algebraic K-theory group in degree d of the algebra of smooth functions on X to the degree -d-1 topological K-theory of X with coefficients in C/Z. This map generalizes the map used by Suslin in order to calculate the torsion subgroup of algebraic K-theory of C (the case X=*). We state and partially prove a conjecture which compares the composition of the map above with the evaluation against the K-homology class of a Dirac operator on X on the one hand, and the Connes-Karoubi multiplicative character of the associated d-summable Fredholm module on the other.

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arXiv2014

The universal eta-invariant for manifolds with boundary

We extend the theory of the universal eta-invariant to the case of relative bordism groups of manifolds with boundaries. This allows the construction of secondary descendants of the universal eta-invariant. We obtain an interpretation of Laures' f-invariant as an example of this general construction. As an aside we improve a recent result of Han-Zhang on the modularity of a certain power series of eta invariants.

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arXiv2014

On the topological contents of eta invariants

We discuss an universal bordism invariant obtained from the Atiyah-Patodi-Singer eta-invariant from the analytic and homotopy theoretic point of view. Classical invariants like the Adams e-invariant, $ρ$-invariants and $String$-bordism invariants are derived as special cases. The main results are a secondary index theorem about the coincidence of the analytic and topological constructions and intrinsic expressions for the bordism invariants.

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arXiv2014

Twisted differential cohomology

The main goal of the present paper is the construction of twisted generalized differential cohomology theories and the comprehensive statement of its basic functorial properties. Technically it combines the homotopy theoretic approach to (untwisted) generalized differential cohomology developed by Hopkins-Singer and later by the first author and D. Gepner with the oo-categorical treatement of twisted cohomology by Ando-Blumberg-Gepner. We introduce the notion of a differential twist for a given generalized cohomology theory and construct twisted differential cohomology groups (resp. spectra). The main technical results of the paper are existence and uniqueness statements for differential twists. These results will be applied in a variety of examples, including K-theory, topological modular forms and other cohomology theories.

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arXiv2013

Differential cohomology theories as sheaves of spectra

We show that every sheaf on the site of smooth manifolds with values in a stable (infinity,1)-category (like spectra or chain complexes) gives rise to a differential cohomology diagram and a homotopy formula, which are common features of all classical examples of differential cohomology theories. These structures are naturally derived from a canonical decomposition of a sheaf into a homotopy invariant part and a piece which has a trivial evaluation on a point. In the classical examples the latter is the contribution of differential forms. This decomposition suggest a natural scheme to analyse new sheaves by determining these pieces and the gluing data. We perform this analysis for a variety of classical and not so classical examples.

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arXiv2013

Differential cohomology

These course note first provide an introduction to secondary characteristic classes and differential cohomology. They continue with a presentation of a stable homotopy theoretic approach to the theory of differential extensions of generalized cohomology theories including products and Umkehr maps.

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arXiv2012

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.

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arXiv2011

Differential K-theory. A survey

Generalized differential cohomology theories, in particular differential K-theory (often called "smooth K-theory"), are becoming an important tool in differential geometry and in mathematical physics. In this survey, we describe the developments of the recent decades in this area. In particular, we discuss axiomatic characterizations of differential K-theory (and that these uniquely characterize differential K-theory). We describe several explicit constructions, based on vector bundles, on families of differential operators, or using homotopy theory and classifying spaces. We explain the most important properties, in particular about the multiplicative structure and push-forward maps and will state versions of the Riemann-Roch theorem and of Atiyah-Singer family index theorem for differential K-theory.

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arXiv2011

Differential orbifold K-Theory

We construct differential equivariant K-theory of representable smooth orbifolds as a ring valued functor with the usual properties of a differential extension of a cohomology theory. For proper submersions (with smooth fibres) we construct a push-forward map in differential equivariant K-theory. Finally, we construct a non-degenerate intersection pairing with values in C/Z for the subclass of smooth orbifolds which can be written as global quotients by a finite group action. We construct a real subfunctor of our theory, where the pairing restricts to a non-degenerate R/Z-valued pairing.

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