K-theory for the maximal Roe algebra of certain expanders

We study in this paper the maximal version of the coarse Baum-Connes assembly map for families of expanding graphs arising from residually finite groups. Unlike for the usual Roe algebra, we show that this assembly map is closely related to the (maximal) Baum-Connes assembly map for the group and is an isomorphism for a class of expanders. We also introduce a quantitative Baum-Connes assembly map and discuss its connections to K-theory of (maximal) Roe algebras.     

An equivariant index for proper actions III: The invariant and discrete series indices

We study two special cases of the equivariant index defined in part I of this series. We apply this index to deformations of ${\mathrm{Spin}}^{\mathrm{c}}$-Dirac operators, invariant under actions by possibly noncompact groups, with possibly noncompact orbit spaces. One special case is an index defined in terms of multiplicities of discrete series representations of semisimple groups, where we assume the Riemannian metric to have a certain product form. The other is an index defined in terms of sections invariant under a group action. We obtain a relation with the analytic assembly map, quantisation commutes with reduction results, and Atiyah–Hirzebruch type vanishing theorems. The arguments are based on an explicit decomposition of ${\mathrm{Spin}}^{\mathrm{c}}$-Dirac operators with respect to a global slice for the action.     

Witten deformation and the equivariant index

Let M be a compact Riemannian manifold endowed with an isometric action of a compact, connected Lie group. The method of the Witten deformation is used to compute the virtual representation-valued equivariant index of a transversally elliptic, first order differential operator on M. The multiplicities of irreducible representations in the index are expressed in terms of local quantities associated to the isolated singular points of an equivariant bundle map that is locally Clifford multiplication by a Killing vector field near these points.      

The coarse geometric Novikov conjecture for subspaces of non-positively curved manifolds

In this paper, we prove the coarse geometric Novikov conjecture for metric spaces with bounded geometry which admit a coarse embedding into a simply connected complete Riemannian manifold of non-positive sectional curvature.     

Localization algebras and the coarse Baum--Connes conjecture

We introduce a localization algebra to define a local index mapo from K-homology to the K-theory of the localization algebra. we show that the local index map is an isomorphism. We apply this result to study the coarse Baum-Connes conjecture.     

Enlargeability and index theory: infinite covers

In Hanke and Schick (J Differ Geom 74(2):293–320, 2006) we showed non-vanishing of the universal index elements in the K-theory of the maximal C*-algebras of the fundamental groups of enlargeable spin manifolds. The underlying notion of enlargeability was the one from Gromov and Lawson (Ann Math 111(2):209–230, 1980), involving contracting maps defined on finite covers of the given manifolds. In the paper at hand, we weaken this assumption to the one in Gromov and Lawson (Publ IHES 58:83–196, 1983) where infinite covers are allowed. The new idea is the construction of a geometrically given C*-algebra with trace which encodes the information given by these infinite covers. Along the way we obtain an easy proof of a relative index theorem relevant in this context. We thank S. Stolz and A. Thom for useful conversations regarding the research in this paper. Both authors are members of the DFG emphasis programme “Globale Differentialgeometrie” whose support is gratefully acknowledged.     

The coarse geometric Novikov conjecture and uniform convexity

The coarse geometric Novikov conjecture provides an algorithm to determine when the higher index of an elliptic operator on a noncompact space is nonzero. The purpose of this paper is to prove the coarse geometric Novikov conjecture for spaces which admit a (coarse) uniform embedding into a uniformly convex Banach space.     

An index theory for countably P-concentrative J maps

In this paper we present a fixed-point index for countably P-concentrative J maps.     

The Novikov Conjecture for Low Degree Cohomology Classes

We outline a twisted analogue of the Mishchenko–Kasparov approach to prove the Novikov conjecture on the homotopy invariance of the higher signatures. Using our approach, we give a new and simple proof of the homotopy invariance of the higher signatures associated to all cohomology classes of the classifying space that belong to the subring of the cohomology ring of the classifying space that is generated by cohomology classes of degree less than or equal to 2, a result that was first established by Connes and Gromov and Moscovici using other methods. A key new ingredient is the construction of a tautological C* _r (, )-bundle and connection, which can be used to construct a C* _r (, )-index that lies in the Grothendieck group of C* _r (, ), where is a multiplier on the discrete group corresponding to a degree 2 cohomology class. We also utilise a main result of Hilsum and Skandalis to establish our theorem.     

Positive scalar curvature,higher rho invariants and localization algebras

In this paper, we use localization algebras to study higher rho invariants of closed spin manifolds with positive scalar curvature metrics. The higher rho invariant is a secondary invariant and is closely related to positive scalar curvature problems. The main result of the paper connects the higher index of the Dirac operator on a spin manifold with boundary to the higher rho invariant of the Dirac operator on the boundary, where the boundary is endowed with a positive scalar curvature metric. Our result extends a theorem of Piazza and Schick [27, Theorem 1.17].