Higson Corona Research Articles

Small subsets of asymptotic resemblance spaces and their Higson corona

Small subsets of asymptotic resemblance spaces and their Higson corona

Twisted coefficients on coarse spaces and their corona

To a metric space X we associate a compact topological space ν′(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u '({X})$$\\end{document} called the corona of X. Then a coarse map f:X→Y\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:X\\rightarrow Y$$\\end{document} between metric spaces is mapped to a continuous map ν′(f):ν′(X)→ν′(Y)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u '({f}):\ u '({X})\\rightarrow \ u '({Y})$$\\end{document} between coronas. Sheaf cohomology assigned to a coarse metric space is preserved and reflected by the corona functor. This work reveals new tools to analyze the Higson corona.

Coarse Sheaf Cohomology

A certain Grothendieck topology assigned to a metric space gives rise to a sheaf cohomology theory which sees the coarse structure of the space. Already constant coefficients produce interesting cohomology groups. In degree 0, they see the number of ends of the space. In this paper, a resolution of the constant sheaf via cochains is developed. It serves to be a valuable tool for computing cohomology. In addition, coarse homotopy invariance of coarse cohomology with constant coefficients is established. This property can be used to compute cohomology of Riemannian manifolds. The Higson corona of a proper metric space is shown to reflect sheaves and sheaf cohomology. Thus, we can use topological tools on compact Hausdorff spaces in our computations. In particular, if the asymptotic dimension of a proper metric space is finite, then higher cohomology groups vanish. We compute a few examples. As it turns out, finite abelian groups are best suited as coefficients on finitely generated groups.

Slant products on the Higson–Roe exact sequence

We construct a slant product $/ \colon \mathrm{S}_p(X \times Y) \times \mathrm{K}_{1-q}(\mathfrak{c}^{\mathrm{red}}Y) \to \mathrm{S}_{p-q}(X)$ on the analytic structure group of Higson and Roe and the K-theory of the stable Higson corona of Emerson and Meyer. The latter is the domain of the co-assembly map $\mu^\ast \colon \mathrm{K}_{1-\ast}(\mathfrak{c}^{\mathrm{red}}Y) \to \mathrm{K}^\ast(Y)$. We obtain such products on the entire Higson--Roe sequence. They imply injectivity results for external product maps. Our results apply to products with aspherical manifolds whose fundamental groups admit coarse embeddings into Hilbert space. To conceptualize the class of manifolds where this method applies, we say that a complete $\mathrm{spin}^{\mathrm{c}}$-manifold is Higson-essential if its fundamental class is detected by the co-assembly map. We prove that coarsely hypereuclidean manifolds are Higson-essential. We draw conclusions for positive scalar curvature metrics on product spaces, particularly on non-compact manifolds. We also obtain equivariant versions of our constructions and discuss related problems of exactness and amenability of the stable Higson corona.

Perfectness of Higson type compactifications

In this paper, we study perfectness of Higson type compactifications and give a necessary and sufficient condition that the subpower and the sublinear Higson compactification of a locally connected proper metric space are perfect. Consequently, we show that the sublinear Higson corona of the half open interval with the usual metric is an indecomposable continuum which is an affirmative answer to Iwamoto's question [9].

Coarse homotopy on metric spaces and their corona

This paper discusses properties of the Higson corona by means of a quotient on coarse ultrafilters on a proper metric space. We use this description to show that the corona functor is faithful and reflects isomorphisms.

Secondary cup and cap products in coarse geometry

We construct secondary cup and cap products on coarse (co-)homology theories from given cross and slant products. They are defined for coarse spaces relative to weak generalized controlled deformation retracts. On ordinary coarse cohomology, our secondary cup product agrees with a secondary product defined by Roe. For coarsifications of topological coarse (co-)homology theories, our secondary cup and cap products correspond to the primary cup and cap products on Higson dominated coronas via transgression maps. And in the case of coarse mathrm {K}-theory and -homology, the secondary products correspond to canonical primary products between the mathrm {K}-theories of the stable Higson corona and the Roe algebra under assembly and co-assembly.

Connectedness properties of the Higson corona of the half line

It is proved that the Higson corona of the half line satisfies the following connectedness properties: The Higson corona of the half line is hereditarily unicoherent; under the Continuum Hypothesis, there are 2c composants of the Higson corona of the half line; the principle of Near Coherence Filters is equivalent to the assertion that the Higson corona of the half line has only one composant; no non-degenerate subcontinuum of the Higson corona of the half line is hereditarily indecomposable; and every non-empty open subset of the Higson corona of the half line contains a non-degenerate indecomposable continuum.

Metrics on Doubles as an Inverse Semigroup

For a metric space X we study metrics on the two copies of X. We define composition of such metrics and show that the equivalence classes of metrics are a semigroup M(X). Our main result is that M(X) is an inverse semigroup. Therefore, one can define the \(C^*\)-algebra of this inverse semigroup, which is not necessarily commutative. If the Gromov–Hausdorff distance between two metric spaces, X and Y, is finite then their inverse semigroups M(X) and M(Y) (and hence their \(C^*\)-algebras) are isomorphic. We characterize the metrics that are idempotents, and give examples of metric spaces for which the semigroup M(X) (and the corresponding \(C^*\)-algebra) is commutative. We also describe the class of metrics determined by subsets of X in terms of the closures of the subsets in the Higson corona of X and the class of invertible metrics.

Constructing a coarse space with a given Higson or binary corona

For any compact Hausdorff space K we construct a canonical finitary coarse structure EX,K on the set X of isolated points of K. This construction has two properties:(1)If a finitary coarse space (X,E) is metrizable, then its coarse structure E coincides with the coarse structure EX,K generated by the Higson compactification K of X.(2)A compact Hausdorff space K coincides with the Higson compactification of the coarse space (X,EX,K) if the set X is dense in K and the space K is Fréchet-Urysohn. This implies that a compact Hausdorff space K is homeomorphic to the Higson corona of some finitary coarse space if one of the following conditions holds: (i) K is perfectly normal; (ii) K has weight w(K)≤ω1 and character χ(K)<p. Under CH every (zero-dimensional) compact Hausdorff space of weight ≤ω1 is homeomorphic to the Higson (resp. binary) corona of some cellular finitary coarse space.

Indecomposable continua as Higson coronae

In this paper, we consider spaces whose Higson coronae are indecomposable continua. We show that for a non-compact proper metric space X which is coarsely geodesic and has coarse bounded geometry, the Higson corona of X is an indecomposable continuum if and only if X is coarsely equivalent to the space of natural numbers. Then we give characterizations of finitely generated groups that have one or two ends by decomposability/indecomposability of the components of their Higson coronae.

On convergence in the subpower Higson corona of metric spaces

The subpower Higson corona of a proper metric space is defined by the authors as an analogue of the sublinear Higson corona. We prove that, unlikely to the Higson corona, the closure of a σ-compact subset of the subpower Higson corona of a proper unbounded metric space does not necessarily coincide with the remainder of its Stone-Čech compactification.

Coarse indices of twisted operators

Several formulas for computing coarse indices of twisted Dirac type operators are introduced. One type of such formulas is by composition product in [Formula: see text]-theory. The other type is by module multiplications in [Formula: see text]-theory, which also yields an index theoretic interpretation of the duality between Roe algebra and stable Higson corona.

Ends and Simple Coarse Structures

This paper is devoted to introducing coarse structures in a very simple way, namely as an equivalence relation on the set of simple ends. As an application we show that Gromov boundary of every hyperbolic space is an example of a Higson corona and each Freudenthal compactification is an example of a Higson compactification.

Index of Equivariant Callias-Type Operators and Invariant Metrics of Positive Scalar Curvature

We formulate, for any Lie group G acting isometrically on a manifold M, the general notion of a G-equivariant elliptic operator that is invertible outside of a G-cocompact subset of M. We prove a version of the Rellich lemma for this setting and use this to define the equivariant index of such operators. We show that G-equivariant Callias-type operators are self-adjoint, regular, and hence equivariantly invertible at infinity. Such operators explicitly arise from a pairing of the Dirac operator with the equivariant Higson corona. We apply the theory developed herein to obtain an obstruction to positive scalar curvature metrics on non-cocompact manifolds.

On balanced coronas of groups

Let G be an infinite group, κ be an infinite cardinal, κ≤|G| and let Eκ denote a coarse structure on G with the base {{(x,y):y∈FxF}:F∈[G]<κ}. We prove that if either κ<|G| or κ=|G| and κ is singular then the Higson's corona νκ(G) of the coarse space (G,Eκ) is a singleton. If κ=|G| and κ is regular then νκ(G) contains a copy of the space Uκ of κ-uniform ultrafilters on κ.

An indecomposable continuum as subpower Higson corona

In this paper, we study topological properties of the subpower Higson coronas of proper metric spaces and show that the subpower Higson corona of the half open interval with the usual metric is an indecomposable continuum. Continuous surjections from Higson-type coronas onto a Higson-type compactifications of the half open interval are also constructed.

Monotone-light factorizations in coarse geometry

We introduce large scale analogues of topological monotone and light maps, which we call coarsely monotone and coarsely light maps respectively. We show that these two classes of maps constitute a factorization system on the coarse category. We also show how coarsely monotone maps arise from a reflection in a similar way to classically monotone maps, and prove that coarsely monotone maps are stable under those pullbacks which exist in the coarse category. For the case of maps between proper metric spaces, we exhibit some connections between the coarse and classical notions of monotone and light using the Higson corona. Finally, we look at some coarse properties which are preserved by coarsely light maps such as finite asymptotic dimension and exactness, and make some remarks on the situation for groups and group homomorphisms.

Higson compactification and dimension raising

Let X and Y be proper metric spaces. We show that a coarsely n-to-1 map f:X→Y induces an n-to-1 map of Higson coronas. This viewpoint turns out to be successful in showing that the classical dimension raising theorems hold in large scale; that is, if f:X→Y is a coarsely n-to-1 map between proper metric spaces X and Y then asdim(Y)≤asdim(X)+n−1. Furthermore we introduce coarsely open coarsely n-to-1 maps, which include the natural quotient maps via a finite group action, and prove that they preserve the asymptotic dimension.

Large scale absolute extensors

This paper is devoted to dualization of dimension-theoretical results from the small scale to the large scale. So far there are two approaches for such dualization: one consisting of creating analogs of small scale concepts using covers and the other amounting to the covering dimension of the Higson corona ν(X) of X. The first approach was used by M. Gromov when defining the asymptotic dimension asdim(X) of metric spaces X. The second approach was implicitly contained in the paper [6] by Dranishnikov on asymptotic topology. It is not known if the two approaches yield the same concept. However, Dranishnikov–Keesling–Uspenskiy proved dim(ν(X))≤asdim(X) and Dranishnikov established that dim(ν(X))=asdim(X) provided asdim(X)<∞. We characterize asymptotic dimension (for spaces of finite asymptotic dimension) in terms of extensions of slowly oscillating functions to spheres. Our approach is specifically designed to relate asymptotic dimension to the covering dimension of the Higson corona ν(X) in case of proper metric spaces X. As an application, we recover the results of Dranishnikov–Keesling–Uspenskiy and Dranishnikov.