arXiv research

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 …

We introduce the concept of Roe C*-algebra for a locally compact groupoid whose unit space is in general not compact, and that is equipped with an appropriate coarse structure and Haar system. Using Connes' tangent groupoid method, we introduce an analytic index for an elliptic differential operator on a Lie groupoid e…

Let $M$ be a complete Riemannian manifold and assume that $M$ is partitioned by a hypersurface $N$. In this paper we introduce a novel class of functions $C_{\mathrm{w}}(M)$ on noncompact manifolds, which is slightly larger than the algebra of Higson functions. Out of $φ$ that belongs to $C_{\mathrm{w}}(M)$ we construc…

We show that if a countable discrete group acts properly and isometrically on a spin manifold of bounded Riemannian geometry and uniformly positive scalar curvature, then, under a suitable condition on the group action, the maximal higher index of the Dirac operator vanishes in K-theory of the maximal equivariant Roe a…

Defines new Roe algebras for cylindrical spaces, solving metric curvature problems.

Generalizes Roe's theorem to noncompact hypersurfaces.

For a discrete metric space (or more generally a large scale space) $X$ and an action of a group $G$ on $X$ by coarse equivalences, we define a type of coarse quotient space $X_G$, which agrees up to coarse equivalence with the orbit space $X/G$ when $G$ is finite. We then restrict our attention to what we call coarsel…

We will show that for a polynomially contractible manifold of bounded geometry and of polynomial volume growth every coarse and rough cohomology class pairs continuously with the K-theory of the uniform Roe algebra. As an application we will discuss non-vanishing of rough index classes of Dirac operators over such mani…

Following Roe and others (see, e.g., [MR1451755]), we (re)develop coarse geometry from the foundations, taking a categorical point of view. In this paper, we concentrate on the discrete case in which topology plays no role. Our theory is particularly suited to the development of the_Roe (C*-)algebras_ C*(X) and their K…

Formula for fixed points on noncompact spaces.

The thesis defines and proves invariants for manifolds of bounded geometry.

Several formulas for computing coarse indices of twisted Dirac type operators are introduced. One type of such formulas is by composition product in $E$-theory. The other type is by module multiplications in $K$-theory, which also yields an index theoretic interpretation of the duality between Roe algebra and stable Hi…

Uniformly proves index invariance for signature operators on manifolds.

In work of Higson-Roe the fundamental role of the signature as a homotopy and bordism invariant for oriented manifolds is made manifest in how it and related secondary invariants define a natural transformation between the (Browder-Novikov-Sullivan-Wall) surgery exact sequence and a long exact sequence of C*-algebra K-…

New theorem shows gaps in magnetic Schrödinger operator spectra for large coupling.

We study a two dimensional analogue of the Roe-Higson index theorem for a partitioned manifold. We prove that Connes' pairing of some invertible element with Roe's cyclic one-cocycle coincides to the Fredholm index of a Toeplitz operator. In the proof of this paper, we use some properties of a circle and use Higson's a…

Block and Weinberger show that an arithmetic manifold can be endowed with a positive scalar curvature metric if and only if its $\rationals$-rank exceeds 2. We show in this article that these metrics are never in the same coarse class as the natural metric inherited from the base Lie group. Furthering the coarse $C^\as…

Study evaluates profitability of Islamic banks in Bangladesh using ROA, ROE, and ROD.

A semicontinuous semifinite trace is constructed on the C*-algebra generated by the finite propagation operators acting on the L^2-sections of a hermitian vector bundle on an amenable open manifold of bounded geometry. This trace is the semicontinuous regularization of a functional already considered by J. Roe. As an a…

This paper extends the C*-signature to non-Witt spaces using noncommutative geometric methods.

GenAI adoption paradoxically lowers ROE for U.S. banks, with spillovers but systemic risk concerns.

The equivariant coarse index is well-understood and widely used for actions by discrete groups. We extend the definition of this index to general locally compact groups. We use a suitable notion of admissible modules over $C^*$-algebras of continuous functions to obtain a meaningful index. Inspired by work by Roe, we t…

We generalize Roe's Index Theorem for operators of Dirac type on open manifolds to elliptic pseudodifferential operators. To this end we introduce a class of pseudodifferential operators on manifolds of bounded geometry which is more general than similar classes defined by other authors. We revisit Spakula's uniform K-…

Abstract machinery finds obstructions to uniform positive scalar curvature.

Study examines credit risk's impact on Vietnamese banks' financial performance.

Consider a proper, isometric action by a unimodular, locally compact group $G$ on a complete Riemannian manifold $M$. For equivariant elliptic operators that are invertible outside a cocompact subset of $M$, we show that a localised index in the $K$-theory of the maximal group $C^*$-algebra of $G$ is well-defined. The …

In this paper we prove the existence of a natural mapping from the surgery exact sequence for topological manifolds to the analytic surgery exact sequence of N. Higson and J. Roe. This generalizes the fundamental result of Higson and Roe, but in the treatment given by Piazza and Schick, from smooth manifolds to topolog…

We generalize Roe's index theorem for graded generalized Dirac operators on amenable manifolds to multigraded elliptic uniform pseudodifferential operators. The generalization will follow from a local index theorem that is valid on any manifold of bounded geometry. This local formula incorporates the uniform estimates …

The aim of this paper is to provide a proof for a version of Morse inequality for manifolds with boundary. Our main results are certainly known to the experts on Morse theory, nevertheless it seems necessary to write down a complete proof for it. Our proof is analytic and is based on J. Roe's account of Witten's approa…

In this paper, we extend Roe's cyclic $1$-cocycle to relative settings. We also prove two relative index theorems for partitioned manifolds by using its cyclic cocycle, which are generalizations of index theorems on partitioned manifolds. One of these theorems is a variant of [M. Karami-A.H.S. Sadegh-M.E. Zadeh, arXiv:…

Let $Γ$ be a f.g. discrete group and let $\tilde M$ be a Galois $Γ$-covering of a smooth closed manifold $M$. Let $S_*^Γ(\tilde{M})$ be the analytic structure group, appearing in the Higson-Roe analytic surgery sequence $\to S_*^Γ(\tilde M)\to K_*(M)\to K_*(C_r^*Γ)\to$. We prove that for an arbitrary discrete group $Γ$…

Working with group homomorphisms, a construction of manifolds is introduced to preserve homology groups. The construction gives as special cases Qullien's plus construction with handles obtained by Hausmann, the existence of one-sided $h$-cobordism of Guilbault and Tinsley, the existence of homology spheres and higher-…

The coarse category was established by Roe to distill the salient features of the large-scale approach to metric spaces and groups that was started by Gromov. In this paper, we use the language of coarse spaces to define coarse versions of asymptotic property C and decomposition complexity. We prove that coarse propert…

We introduce a construction adding low-dimensional cells to a space that satisfies certain low-dimensional conditions; it preserves high-dimensional homology with appropriate coefficients. This includes as special cases Quillen's plus construction, Bousfield's integral homology localization, the existence of Moore spac…

This thesis studies positive scalar curvature metrics on G-proper spaces and pseudomanifolds.

We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension $\as_{\Z} X$ of metric spaces. We show that it agrees with the asymptotic dimension $\as X$ when the later is finite. Then we use this fact to construct an example of a metric space $X$ of boun…

We prove a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle with boundary; in particular, we define a Godbillon-Vey eta invariant on the boundary-foliation; this is a secondary invariant for longitudinal Dirac operators on type-III foliations. Moreover, employing the Godbillon-Vey index…

The main result of this paper is a new and direct proof of the natural transformation from the surgery exact sequence in topology to the analytic K-theory sequence of Higson and Roe. Our approach makes crucial use of analytic properties and new index theorems for the signature operator on Galois coverings with boundary…

In this paper we state and prove a higher index theorem for an odd-dimensional connected spin riemannian manifold $(M,g)$ which is partitioned by an oriented closed hypersurface $N$. This index theorem generalizes a theorem due to N. Higson and J. Roe in the context of Hilbert modules. Then we apply this theorem to pro…

Defines coarse cohomology of space complements, proving new duality results.

A discrete time probabilistic model, for optimal equity allocation and portfolio selection, is formulated so as to apply to (at least) reinsurance. In the context of a company with several portfolios (or subsidiaries), representing both liabilities and assets, it is proved that the model has solutions respecting constr…

In this paper, we study the space of metrics of positive scalar curvature using methods from coarse geometry. Given a closed spin manifold M with fundamental group G, Stephan Stolz introduced the positive scalar curvature exact sequence, in analogy to the surgery exact sequence in topology. It calculates a structure gr…

In this paper, we investigate analytical and geometric properties of certain non-compact boundary-manifolds, namely manifolds of bounded geometry. One result are strong Bochner type vanishing results for the L^2-cohomology of these manifolds: if e.g. a manifold admits a metric of bounded geometry which outside a compac…

We obtain two types of results on positive scalar curvature metrics for compact spin manifolds that are even dimensional. The first type of result are obstructions to the existence of positive scalar curvature metrics on such manifolds, expressed in terms of end-periodic eta invariants that were defined by Mrowka-Ruber…

We derive a general obstruction to the existence of Riemannian metrics of positive scalar curvature on closed spin manifolds in terms of hypersurfaces of codimension two. The proof is based on coarse index theory for Dirac operators that are twisted with Hilbert C*-module bundles. Along the way we give a complete and s…

Paper constructs a transfer map for codimension 2 submanifolds in higher index theory.

Using ideas from shape theory we embed the coarse category of metric spaces into the category of direct sequences of simplicial complexes with bonding maps being simplicial. Two direct sequences of simplicial complexes are equivalent if one of them can be transformed to the other by contiguous factorizations of bonding…

We had previously defined the rho invariant $ρ_{spin}(Y,E,H, g)$ for the twisted Dirac operator $\not\partial^E_H$ on a closed odd dimensional Riemannian spin manifold $(Y, g)$, acting on sections of a flat hermitian vector bundle $E$ over $Y$, where $H = \sum i^{j+1} H_{2j+1}$ is an odd-degree differential form on $Y…