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.     

The coarse Baum–Connes conjecture via coarse geometry

The C₀ coarse structure on a metric space is a refinement of the bounded structure and is closely related to the topology of the space. In this paper we will prove the C₀ version of the coarse Baum–Connes conjecture and show that K_*(C^*X₀) is a topological invariant for a broad class of metric spaces. Using this result we construct a ‘geometric’ obstruction group to the coarse Baum–Connes conjecture for the bounded coarse structure. We then show under the assumption of finite asymptotic dimension that the obstructions vanish, and hence we obtain a new proof of the coarse Baum–Connes conjecture in this context.     

Hyperbolic metric spaces and the exotic cohomology Novikov conjecture

A geometric version of the Novikov conjecture states that certain cohomology classes of a complete metric space arise from an ideal boundary. We prove this for spaces hyperbolic in the sense of Gromov.     

La conjecture de Novikov pour les feuilletages hyperboliques

Nous définissons la notion de bolicité pour les feuilletages, qui est une notion plus faible que l'hyperbolicité de Gromov, et nous démontrons la conjecture de Novikov pour les feuilletages boliques à base compacte dont le groupoï de d'holonomie est séparé en établissant l'injectivité de l'application de Baum–Connes. Ce résultat généralise celui de Kasparov et Skandalis obtenu dans le cas des groupes boliques.We define the notion of bolicity for foliations, which is a weaker notion than Gromov's hyperbolicity, and we prove the Novikov conjecture for foliations with compact base and whose holonomy groupoid is Hausdorff, by showing that the Baum–Connes map is injective. This result generalizes that of Kasparov and Skandalis in the case of bolic groups.     

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.     

A note on the Osserman conjecture and isotropic covariant derivative of curvature

Let be a Riemannian manifold with the Jacobi operator, which has constant eigenvalues, independent on the unit vector and the point . Osserman conjectured that these manifolds are flat or rank-one locally symmetric spaces (). It is known that for a general pseudo-Riemannian manifold, the Osserman-type conjecture is not true and 4-dimensional Kleinian Jordan-Osserman manifolds are curvature homogeneous. We show that the length of the first covariant derivative of the curvature tensor is isotropic, i.e. . For known examples of 4-dimensional Osserman manifolds of signature we check also that . By the presentation of a class of examples we show that curvature homogeneity and do not imply local homogeneity; in contrast to the situation in the Riemannian geometry, where it is unknown if the Osserman condition implies local homogeneity.

     

Metrics with Positive Scalar Curvature at Infinity and Localization Algebra

In this paper, the authors give a new proof of Block and Weinberger’s Bochner vanishing theorem built on direct computations in the K-theory of the localization algebra.