Sublinear Higson corona of Euclidean cone

Abstract

Let X be a proper metric space. The sublinear Higson compactification h_LX is a variant of the Higson compactification. Its boundary h_LX\X is denoted ν_LX, and is called the sublinear Higson corona of X. The sublinear Higson corona is a functor from the category of coarse spaces to that of compact Hausdorff spaces. Let P be a compact metric space and X be an unbounded proper metric space. We show that the sublinear Higson corona of a product space P × X equipped with a cone metric is homeomorphic to a product P × ν_LX. Especially, the sublinear Higson corona of the n-dimensional Euclidean space is homeomorphic to the product of an (n − 1)-dimensional sphere and the sublinear Higson corona of natural numbers.