arXiv subjects

Charlotte Wahl

Publications and source records attributed to Charlotte Wahl.

13 recordsLinked to original sources

arXiv2015

Two-cocycle twists and Atiyah-Patodi-Singer index theory

We construct eta- and rho-invariants for Dirac operators, on the universal covering of a closed manifold, that are invariant under the projective action associated to a 2-cocycle of the fundamental group. We prove an Atiyah-Patodi-Singer index theorem in this setting, as well as its higher generalization. Applications concern the classification of positive scalar curvature metrics on closed spin manifolds. We also investigate the properties of these twisted invariants for the signature operator and the relation to the higher invariants.

math.DG↗

arXiv2014

An interesting example for spectral invariants

In "Illinois J. of Math. {\bf 38} (1994) 653--678", the heat operator of a Bismut superconnection for a family of generalized Dirac operators is defined along the leaves of a foliation with Hausdorff groupoid. The Novikov-Shubin invariants of the Dirac operators were assumed greater than three times the codimension of the foliation. It was then showed that the associated heat operator converges to the Chern character of the index bundle of the operator. In "J. K-Theory {\bf 1} (2008) 305--356", we improved this result by reducing the requirement on the Novikov-Shubin invariants to one half of the codimension. In this paper, we construct examples which show that this is the best possible result.

math.GT↗

arXiv2012

Higher rho-invariants and the surgery structure set

We study noncommutative eta- and rho-forms for homotopy equivalences. We prove a product formula for them and show that the rho-forms are well-defined on the structure set. We also define an index theoretic map from L-theory to C*-algebraic K-theory and show that it is compatible with the rho-forms. Our approach, which is based on methods of Hilsum-Skandalis and Piazza-Schick, also yields a unified analytic proof of the homotopy invariance of the higher signature class and of the L^2-signature for manifolds with boundary.

math.DG↗

arXiv2012

The Atiyah-Patodi-Singer index theorem for Dirac operators over C*-algebras

We prove an Atiyah-Patodi-Singer index theorem for Dirac operators twisted by C*-vector bundles. We use it to derive a general product formula for eta-forms and to define and study new rho-invariants generalizing Lott's higher rho-form. The higher Atiyah-Patodi-Singer index theorem of Leichtnam-Piazza can be recovered by applying the theorem to Dirac operators twisted by the Mishenko-Fomenko bundle associated to the reduced C*-algebra of the fundamental group.

math.DG↗

arXiv2009

Spectral flow, index and the signature operator

We relate the spectral flow to the index for paths of selfadjoint Breuer-Fredholm operators affiliated to a semifinite von Neumann algebra, generalizing results of Robbin-Salamon and Pushnitski. Then we prove the vanishing of the von Neumann spectral flow for the tangential signature operator of a foliated manifold when the metric is varied. We conclude that the tangential signature of a foliated manifold with boundary does not depend on the metric. In the Appendix we reconsider integral formulas for the spectral flow of paths of bounded operators.

math.DG↗

arXiv2009

Product formula for Atiyah-Patodi-Singer index classes and higher signatures

We define generalized Atiyah-Patodi-Singer boundary conditions of product type for Dirac operators associated to C*-vector bundles on the product of a compact manifold with boundary and a closed manifold. We prove a product formula for the K-theoretic index classes, which we use to generalize the product formula for the topological signature to higher signatures.

math.DG↗

arXiv2009

Homological index formulas for elliptic operators over C*-algebras

We prove index formulas for elliptic operators acting between sections of C*-vector bundles on a closed manifold. The formulas involve Karoubi's Chern character from K-theory of a C*-algebra to de Rham homology of smooth subalgebras. We show how they apply to the higher index theorem for coverings and to flat foliated bundles, and prove an index theorem for C*-dynamical systems associated to actions of compact Lie groups. In an Appendix we relate the pairing of odd K-theory and KK-theory to the noncommutative spectral flow and prove the regularity of elliptic pseudodifferential operators over C*-algebras.

math.KT↗

arXiv2008

Index theory for actions of compact Lie groups on C*-algebras

We study the index theory for actions of compact Lie groups on C*-algebras with an emphasis on principal actions. Given an invariant semifinite trace on the C*-algebra we obtain semifinite spectral triples. For circle actions we consider the relation to the dual Pimsner-Voiculescu sequence. On the way we show that the notions ``saturated'' and ``principal'' are equivalent for actions by compact Lie groups.

math.KT↗

arXiv2007

On the noncommutative spectral flow

We define and study the noncommutative spectral flow for paths of regular selfadjoint Fredholm operators on a countably generated Hilbert C*-module. We give an axiomatic description and discuss some applications. One of them is the definition of a noncommutative Maslov index for paths of Lagrangians which appears in a splitting formula for the spectral flow. Analogously we study the spectral flow for odd operators on a graded module.

math.OA↗

arXiv2007

A new topology on the space of unbounded selfadjoint operators and the spectral flow

We define a new topology, weaker than the gap topology, on the space of selfadjoint unbounded operators on a separable Hilbert space. We show that the subspace of selfadjoint Fredholm operators represents the functor $K^1$ from the category of compact spaces to the category of abelian groups and prove a similar result for $K^0$. We define the spectral flow of a continuous path of selfadjoint Fredholm operators generalizing the approach of Booss-Bavnek--Lesch--Phillips.

math.FA↗

arXiv2007

Spectral flow and winding number in von Neumann algebras

We introduce a new topology, weaker than the gap topology, on the space of selfadjoint operators affiliated to a semifinite von Neumann algebra. We define the real-valued spectral flow for a continuous path of selfadjoint Breuer-Fredholm operators in terms of a generalization of the winding number. We compare our definition with Phillips' analytical definition and derive integral formulas for the spectral flow for certain paths of unbounded operators with common domain, generalizing those of Carey--Phillips. Furthermore we prove the homotopy invariance of the real-valued index. As an example we consider invariant symmetric elliptic differential operators on Galois coverings.

math.OA↗

arXiv2007

Spectral flow as winding number and integral formulas

A general integral formula for the spectral flow of a path of unbounded selfadjoint Fredholm operators subject to certain summability conditions is derived from the interpretation of the spectral flow as a winding number.

math.FA↗

arXiv2006

Noncommutative Maslov Index and Eta Forms

We define and prove a noncommutative generalization of a formula relating the Maslov index of a triple of Lagrangian subspaces of a symplectic vector space to eta-invariants associated to a pair of Lagrangian subspaces generalizing a result of Bunke and Koch in the family case. The noncommutative Maslov index, defined for modules over a $C^*$-algebra $\A$, is an element in $K_0(\A)$. The generalized formula calculates its Chern character in the de Rham homology of certain dense subalgebras of $\A$. The proof, modelled on the proof by Bunke and Koch, is a noncommutative Atiyah-Patodi-Singer index theorem for a particular Dirac operator twisted by an $\A$-vector bundle. We develop an analytic framework for this type of index problem.

math.KT↗