A combinatorial approach to coarse geometry
Ordinary shape theory uses mappings of metric spaces \((X,d_Y)\) into CW complexes. The present authors deal with the coarse category of metric spaces, where homotopy of maps \(f,g: X \longrightarrow (Y,d_Y) \), is replaced by \textit{large scale} (\(ls-\)) equivalent functions: \(f \sim_{ls} g\) whenever \(sup_{x \in X} (d_Y (f(x),g(x)) < \infty))\). The mappings in this category are called \textit{bornologous} functions. Now the authors construct \textit{coarse simplicial complexes} consisting of sequences \(K_1 \longrightarrow K_2 \longrightarrow \dots\) having special properties. In analogy to shape theory there are coarse complexes associated with a metric space, satisfying conditions analog to those of a Čech system in shape theory. As a theorem they have:NEWLINENEWLINE If \((X,d_X), \;(Y,d_Y))\) are metric spaces with coarse complexes \(\mathcal{K}\), resp. \(\mathcal{L}\), then there is a natural bijection between bornologogous maps from \(X\) to \(Y\) and the set of morphisms from \(\mathcal{K}\) to \(\mathcal{L}\).NEWLINENEWLINESince here complexes are mapped into spaces, the author recognized a duality to shape theory, where this works the other way. In addition a concept of \textit{coarse} dimension, connectedness and some property A is treated.
scientific article; zbMATH DE number 6023679
scientific article