184 research outputs found
Asymptotic topology
We establish some basic theorems in dimension theory and absolute extensor
theory in the coarse category of metric spaces. Some of the statements in this
category can be translated in general topology language by applying the Higson
corona functor. The relation of problems and results of this `Asymptotic
Topology' to Novikov and similar conjectures is discussed.Comment: 38 pages, AMSTe
Coarse equivalence versus bijective coarse equivalence of expander graphs
We provide a characterization of when a coarse equivalence between coarse
disjoint unions of expander graphs is close to a bijective coarse equivalence.
We use this to show that if the uniform Roe algebras of coarse disjoint unions
of expanders graphs are isomorphic, then the metric spaces must be bijectively
coarsely equivalent
Slant products on the Higson-Roe exact sequence
We construct a slant product 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 . 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
-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.Comment: 82 pages; v2: Minor improvements. To appear in Ann. Inst. Fourie
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