scholarly journals Some ‘homological’ properties of the stable Higson corona

2013 ◽  
Vol 7 (1) ◽  
pp. 203-220
Author(s):  
Rufus Willett
Keyword(s):  
Higson Corona

scholarly journals Subcontinua of the Higson corona

1997 ◽  
Vol 80 (1-2) ◽  
pp. 155-160 ◽  
Author(s):  
James Keesling
Keyword(s):  
Higson Corona

scholarly journals Coarse indices of twisted operators

2019 ◽  
Vol 11 (04) ◽  
pp. 823-873
Author(s):  
Christopher Wulff
Keyword(s):  
The Other ◽  
Dirac Type ◽  
Roe Algebra ◽  
Composition Product ◽  
Higson Corona

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.

scholarly journals Coarse dynamics and fixed-point theorem

2011 ◽  
Vol 202 ◽  
pp. 1-13
Author(s):  
Tomohiro Fukaya
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Point Of View ◽  
Abelian Semigroup ◽  
Semigroup Actions ◽  
Brouwer’S Fixed Point Theorem ◽  
Brouwer's Fixed Point Theorem ◽  
Higson Corona ◽  
Coarse Space

AbstractWe study semigroup actions on a coarse space and the induced actions on the Higson corona from a dynamical point of view. Our main theorem states that if an action of an abelian semigroup on a proper coarse space satisfies certain conditions, the induced action has a fixed point in the Higson corona. As a corollary, we deduce a coarse version of Brouwer’s fixed-point theorem.

scholarly journals Metric Compactifications and Coarse Structures

2015 ◽  
Vol 67 (5) ◽  
pp. 1091-1108 ◽  
Author(s):  
Kotaro Mine ◽  
Atsushi Yamashita
Keyword(s):  
Compact Space ◽  
Metric Spaces ◽  
Locally Compact ◽  
Totally Bounded ◽  
Compact Metric Spaces ◽  
Locally Compact Space ◽  
Equivalence Of Categories ◽  
Metrizable Spaces ◽  
Coarse Structures ◽  
Higson Corona

AbstractLet TB be the category of totally bounded, locally compact metric spaces with the C0 coarse structures. We show that if X and Y are in TB, then X and Y are coarsely equivalent if and only if their Higson coronas are homeomorphic. In fact, the Higson corona functor gives an equivalence of categories TB → K, where K is the category of compact metrizable spaces. We use this fact to show that the continuously controlled coarse structure on a locally compact space X induced by some metrizable compactification is determined only by the topology of the remainder .

scholarly journals Secondary cup and cap products in coarse geometry

2021 ◽  
Vol 8 (3) ◽  
Author(s):  
Christopher Wulff
Keyword(s):  
Secondary Product ◽  
Secondary Products ◽  
Coarse Geometry ◽  
Cup Product ◽  
Primary Products ◽  
Roe Algebra ◽  
Higson Corona

AbstractWe 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 $$\mathrm {K}$$ K -theory and -homology, the secondary products correspond to canonical primary products between the $$\mathrm {K}$$ K -theories of the stable Higson corona and the Roe algebra under assembly and co-assembly.

On convergence in the subpower Higson corona of metric spaces

2020 ◽  
Vol 272 ◽  
pp. 107068
Author(s):  
Jacek Kucab ◽  
Mykhailo Zarichnyi
Keyword(s):  
Metric Spaces ◽  
Higson Corona

scholarly journals Coronae of relatively hyperbolic groups and coarse cohomologies

2016 ◽  
Vol 08 (03) ◽  
pp. 431-474 ◽  
Author(s):  
Tomohiro Fukaya ◽  
Shin-ichi Oguni
Keyword(s):  
Hyperbolic Group ◽  
3 Dimensional ◽  
Relatively Hyperbolic Groups ◽  
Negatively Curved ◽  
Blowing Up ◽  
Roe Algebra ◽  
Assembly Map ◽  
Complete Riemannian Manifolds ◽  
Relatively Hyperbolic ◽  
Higson Corona

We construct a corona of a relatively hyperbolic group by blowing-up all parabolic points of its Bowditch boundary. We relate the [Formula: see text]-homology of the corona with the [Formula: see text]-theory of the Roe algebra, via the coarse assembly map. We also establish a dual theory, that is, we relate the [Formula: see text]-theory of the corona with the [Formula: see text]-theory of the reduced stable Higson corona via the coarse co-assembly map. For that purpose, we formulate generalized coarse cohomology theories. As an application, we give an explicit computation of the [Formula: see text]-theory of the Roe-algebra and that of the reduced stable Higson corona of the fundamental groups of closed 3-dimensional manifolds and of pinched negatively curved complete Riemannian manifolds with finite volume.

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