Coarse Category Research Articles

COARSE AMENABILITY AND DISCRETENESS

This paper is devoted to dualization of paracompactness to the coarse category via the concept of $R$-disjointness. Property A of Yu can be seen as a coarse variant of amenability via partitions of unity and leads to a dualization of paracompactness via partitions of unity. On the other hand, finite decomposition complexity of Guentner, Tessera, and Yu and straight finite decomposition complexity of Dranishnikov and Zarichnyi employ $R$-disjointness as the main concept. We generalize both concepts to that of countable asymptotic dimension and our main result shows that it is a subclass of spaces with Property A. In addition, it gives a necessary and sufficient condition for spaces of countable asymptotic dimension to be of finite asymptotic dimension.

Normal sequences in all scales

The metrization theorem of Alexandroff and Urysohn is known as a strong tool in general topology. In this paper, using the notion of Alexandroff and Urysohn metric, we introduce the notions of all-scale normal sequence, large-scale normal sequence, and small-scale normal sequence to establish a new unified approach to study various concepts of metric geometry in all scales. This approach provides us with the interplay between topology and metric geometry. We show that all-scale, large-scale, and small-scale concepts of metric spaces can be studied as topological problems, using the notions of normal sequences in appropriate scales. The concepts which we deal with are metrization, Lipschitz category, the Assouad–Nagata dimension as an all-scale concept; the uniform category, the topological category, the uniform dimension, the covering dimension as a small-scale concept; the coarse category, large-scale Lipschitz maps, quasi-isometric maps, the asymptotic dimension, and the asymptotic Assouad–Nagata dimension as a large-scale concept. This approach also provides us with the interplay between various concepts of metric geometry in different scales. As an application, we obtain some relations of the Assouad–Nagata dimension to the uniform dimension, the covering dimension, and the asymptotic dimension.

A new construction of universal spaces for asymptotic dimension

For each n, we construct a separable metric space Un that is universal in the coarse category of separable metric spaces with asymptotic dimension (asdim) at most n and universal in the uniform category of separable metric spaces with uniform dimension (udim) at most n. Thus, Un serves as a universal space for dimension n in both the large-scale and infinitesimal topology. More precisely, we prove:asdimUn=udimUn=n and such that for each separable metric space X,a)if asdimX⩽n, then X is coarsely equivalent to a subset of Un;b)if udimX⩽n, then X is uniformly homeomorphic to a subset of Un.

Asymptotic dimension, property A, and Lipschitz maps

It is well-known that a paracompact space X is of covering dimension n if and only if any map f from X to a simplicial complex K can be pushed into its n-skeleton. We use the same idea to define dimension in the coarse category. It turns out the analog of maps f from X to K is related to asymptotically Lipschitz maps, the analog of paracompact spaces are spaces related to Yu's Property A, and the dimension coincides with Gromov's asymptotic dimension.

A combinatorial approach to coarse geometry

Using ideas from shape theory we embed the coarse category of metric spaces into the category of direct sequences of simplicial complexes with bonding maps being simplicial. Two direct sequences of simplicial complexes are equivalent if one of them can be transformed to the other by contiguous factorizations of bonding maps and by taking infinite subsequences. This embedding can be realized by either Rips complexes or analogs of Roeʼs anti-Čech approximations of spaces. In this model coarse n-connectedness of K = { K 1 → K 2 → ⋯ } means that for each k there is m > k such that the bonding map from K k to K m induces trivial homomorphisms of all homotopy groups up to and including n. The asymptotic dimension being at most n means that for each k there is m > k such that the bonding map from K k to K m factors (up to contiguity) through an n-dimensional complex. Property A of G. Yu is equivalent to the condition that for each k and for each ϵ > 0 there is m > k such that the bonding map from | K k | to | K m | has a contiguous approximation g : | K k | → | K m | which sends simplices of | K k | to sets of diameter at most ϵ.

$C_{0}$-coarse geometry of complements of Z-sets in the Hilbert cube

Motivated by the Chapman Complement Theorem, we construct an isomorphism between the topological category of compact Z-sets in the Hilbert cube $Q$ and the $C_{0}$-coarse category of their complements. The $C_{0}$-coarse morphisms are, in this particular case, intrinsically related to uniformly continuous proper maps. Using that fact we are able to relate in a natural way some of the topological invariants of Z-sets to the geometry of their complements.

The quantitative analysis of mammographic densities

Quantitative classification of mammographic parenchyma based on radiological assessment has been shown to provide one of the strongest estimates of the risk of developing breast cancer. Existing classification schemes, however, are limited by coarse category scales. In addition, subjectivity can lead to sizeable interobserver and intraobserver variations. Here, the authors propose an interactive thresholding technique applied to digitized film-screen mammograms, which assesses the proportion of the mammographic image representing radiographically dense tissue. Observers viewed images on a CRT display and selected grey-level thresholds from which the breast and regions of dense tissue in the breast were identified. The proportion of radiographic density was then calculated from the image histogram. The technique was evaluated for the mammograms of 30 women and is well correlated (R>0.91, Spearman coefficient) with a six-category subjective classification of radiographic density by radiologists. The technique was found to be very reliable with an intraclass correlation coefficient between observers typically R>0.9. This technique may have a role in routine mammographic analysis for the purpose of assessing risk categories and as a tool in studies of the etiology of breast cancer, in particular for monitoring changes in breast parenchyma during potential preventive interventions.