Asymptotic topology (Q2759252)

This famous paper reveals the deep analogy between usual (local) topology and its global counterpart, called asymptotic topology. The paper consists of 9 sections. In the first section the author introduces the asymptotic category \(\mathcal A\) whose objects are proper metric spaces (that is, metric spaces whose closed bounded subsets are compact) and whose morphisms are proper asymptotically Lipschitz maps. He also introduces two other categories \(\tilde{\mathcal A}\) and \(\bar{\mathcal A}\) differing from \(\mathcal A\) by morphisms.NEWLINENEWLINENEWLINEThe second section describes some basic constructions in the asymptotic categories: product, asymptotic product, quotient spaces, cones, suspensions, the space of probability measures.NEWLINENEWLINENEWLINEThe third section is devoted to absolute extensors in the category \(\mathcal A\). As expected, an object \(Y\) of \(\mathcal A\) is called an absolute extensor in \(\mathcal A\) (denoted by \(Y\in AE(\mathcal A)\)) if any morphism \(\varphi:Z\to Y\) from a closed subset \(Z\) of an object \(X\) of \(\mathcal A\) admits an extension to a morphism \(\bar \varphi:X\to Y\). What is surprising is that the Euclidean space \(\mathbb R^n\) fails to be \(AE(\mathcal A)\) -- in asymptotic topology it plays the role of the \((n-1)\)-dimensional sphere. In contrast, the half-line \(\mathbb R_+\) is an \(AE(\mathcal A)\) and plays the role of a point while its \(n\)-th power \((\mathbb R_+)^n\) is an \(AE(\mathcal A)\), too and plays the role of the \((n-1)\)-dimensional Euclidean space.NEWLINENEWLINENEWLINEIn the fourth section the author studies the notion of an absolute neighborhood extensor and introduces three dimension functions in the category \(\mathcal A\): the Gromov asymptotic dimension \({\text{asdim}}\) (\({\text{asdim}} X\leq n\) if for any \(L>0\) there is a uniformly bounded cover \(\mathcal U\) of \(X\) with Lebesgue number \(>L\) and multiplicity \(\leq n+1\)), the dimension \({\text{asdim}}_*\) (\({\text{asdim}}_*X\leq n\) if for any proper function \(f:X\to\mathbb R_+\) there is a non-expanding map \(\varphi:X\to K\) into an \(n\)-dimensional asymptotic polyhedron such that \(\|\varphi\|<f\); the latter means that for any \(R>0\) the inequality \(\text{diam}(\varphi^{-1}(B_R(\varphi(x))))<f(x)\) holds for all \(x\in X\) outside some compact set), and the dimension \(\dim^c\) related to the Alexandroff-Hurewicz characterization of the covering dimension in terms of extension of maps into spheres (\(\dim^c X\leq n\) if each morphism \(\varphi:A\to\mathbb R^{n+1}\) from a closed subspace of \(X\) can be extended to a morphism \(\bar \varphi:X\to\mathbb R^{n+1}\)). It is shown that \(\dim^cX\leq{\text{ asdim}}_*X\leq{\text{ asdim}} X\) for any object \(X\) of \(\mathcal A\) and all these dimensions are equal if \({\text{ asdim}} X<\infty\).NEWLINENEWLINENEWLINEIn the fifth section with help of the so-called anti-Čech approximation by polyhedra, for a coefficient group \(G\) the asymptotic cohomological dimension \({\text{ asdim}}_GX\) of a proper metric space \(X\) of bounded geometry is introduced and the estimate \({\text{asdim}}_GX\leq{\text{ asdim}} X\) is proved in case of a uniformly contractible space \(X\).NEWLINENEWLINENEWLINEIn the sixth section the author investigates the interplay between the dimension of the Higson corona \(\nu X\) of a proper metric space \(X\) and its asymptotic dimensions. It is shown that \(\dim\nu X=\dim^c X\). Moreover, \(\dim \nu X\) is equal to \({\text{ asdim}} X\) (resp. \({\text{asdim}}_*X\)) whenever the respective asymptotic dimension is finite.NEWLINENEWLINENEWLINEIn the seventh section the author searches for dimensional conditions on a metric space \(X\) guaranteeing that \(X\) admits a coarsely uniform embedding into a Hilbert space. Among such conditions there are \({\text{asdim}}_*X<\infty\), the slow dimension growth and the asymptotic property \(C\) (which means that for any non-decreasing number sequence \((R_n)\) there is a finite sequence of uniformly bounded families \(\{\mathcal U_i\}_{i=1}^k\) of open subsets of \(X\) such that \(\bigcup_{i=1}^k\mathcal U_i=X\) and each family \(\mathcal U_i\) is \(R_i\)-disjoint). In combination with the Yu Theorem (asserting that the coarse Baum-Connes conjecture holds for any uniformly contractible proper metric space \(X\) of bounded geometry that embeds into a Hilbert space) this allows to confirm the Baum-Connes conjecture (and consequently the Novikov conjecture on higher signatures) for proper metric spaces satisfying certain dimension restrictions.NEWLINENEWLINENEWLINEIn the eighth section the author discusses the reduction of the Novikov conjecture on higher signatures to some problems related to Higson coronas.NEWLINENEWLINENEWLINEIn the final ninth section the author poses 14 open problems (some of which are already resolved, see [\textit{M.Zarichnyi}, Asymptotic category and spaces of probability measures, Visnyk Lviv Univ. Ser. Mekh.-Mat. 61, 211-217 (2003)]) and supplies the reader with an amusing micro-macro topological dictionary.