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Register a new userCoarse Homotopy on metric Spaces and their Corona
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Elisa Hartmann
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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. This study provides a Kunneth formula for twisted coarse cohomology. We obtain the Gromov boundary of a hyperbolic proper geodesic metric space as a quotient of its Higson corona.
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To a metric space $X$ we associate a compact topological space $ u X$ called the corona of $X$. Then a coarse map $f:Xto Y$ between metric spaces is mapped to a continuous map $ u f: u Xto u Y$ between coronas. Sheaf cohomology on coarse spaces has been introduced in arXiv:1710.06725. We show the functor $ u$ preserves and reflects sheaf cohomology.
Let $mathfrak{M}$ be a class of metric spaces. A metric space $Y$ is minimal $mathfrak{M}$-universal if every $Xinmathfrak{M}$ can be isometrically embedded in $Y$ but there are no proper subsets of $Y$ satisfying this property. We find conditions under which, for given metric space $X$, there is a class $mathfrak{M}$ of metric spaces such that $X$ is minimal $mathfrak{M}$-universal. We generalize the notion of minimal $mathfrak{M}$-universal metric space to notion of minimal $mathfrak{M}$-universal class of metric spaces and prove the uniqueness, up to an isomorphism, for these classes. The necessary and sufficient conditions under which the disjoint union of the metric spaces belonging to a class $mathfrak{M}$ is minimal $mathfrak{M}$-universal are found. Examples of minimal universal metric spaces are constructed for the classes of the three-point metric spaces and $n$-dimensional normed spaces. Moreover minimal universal metric spaces are found for some subclasses of the class of metric spaces $X$ which possesses the following property. Among every three distinct points of $X$ there is one point lying between the other two points.
We prove metric differentiation for differentiability spaces in the sense of Cheeger. As corollaries we give a new proof that the minimal generalized upper gradient coincides with the pointwise Lipschitz constant for Lipschitz functions on PI spaces, a proof that the Lip-lip constant of any Lip-lip space in the sense of Keith is equal to $1$, and new nonembeddability results.
For a subring $R$ of the rational numbers, we study $R$-localization functors in the local homotopy theory of simplicial presheaves on a small site and then in ${mathbb A}^1$-homotopy theory. To this end, we introduce and analyze two notions of nilpotence for spaces in ${mathbb A}^1$-homotopy theory paying attention to future applications for vector bundles. We show that $R$-localization behaves in a controlled fashion for the nilpotent spaces we consider. We show that the classifying space $BGL_n$ is ${mathbb A}^1$-nilpotent when $n$ is odd, and analyze the (more complicated) situation where $n$ is even as well. We establish analogs of various classical results about rationalization in the context of ${mathbb A}^1$-homotopy theory: if $-1$ is a sum of squares in the base field, ${mathbb A}^n setminus 0$ is rationally equivalent to a suitable motivic Eilenberg--Mac Lane space, and the special linear group decomposes as a product of motivic spheres.
Let (X,d) be a metric space of p-negative type. Recently I. Doust and A. Weston introduced a quantification of the p-negative type property, the so called gap {Gamma} of X. This talk introduces some formulas for the gap {Gamma} of a finite metric space of strict p-negative type and applies them to evaluate {Gamma} for some concrete finite metric spaces.
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