Discrete homology theory for metric spaces (Q2922856)

This paper gives a homological counterpart to the discrete homotopy theory, called \(A\)-theory in honor of \textit{R. Atkin} and his seminal ideas in [``An Algebra of patterns on a complex. I and II'', J. Man-Machine Studies 6, 285--307 (1974), ibid. 8, 448--483 (1976)], whose theoretical foundations where given by \textit{H. Barcelo} et al. [Adv. Appl. Math. 26, No. 2, 97--128 (2001; Zbl 0984.57014)]. As stated in [\textit{E. Babson} et al., J. Algebr. Comb. 24, No. 1, 31--44 (2006; Zbl 1108.05030)], \(A\)-homotopy is a cubical homotopy which applies to graphs (or, more generally and following the ideas of Atkin, to simplicial complexes); this combinatorial homotopy must not be confused with that developed, for instance, by \textit{A. Dochtermann} [Eur. J. Comb. 30, No. 2, 490--509 (2009; Zbl 1167.05017)].NEWLINENEWLINEIn this paper, the authors define a discrete homology theory which is related to \(A\)-homotopy theory in the same way that classical homology is related to classical homotopy theory. Actually, for every real number \(r>0\), by considering metric spaces and \(r\)-Lipschitz functions (i.e. functions \(f:X \to Y\) s. t. \(d_Y(f(x),f(x'))\leq r d_X(x,x')\) for all \(x,x'\) in \(X\)), they generalize the case of graphs and graph homomorphisms (obtained for \(r=1\)). For any metric space \((X,d)\), they get a sequence \(DH_{\star, r}(X)\) of \textit{discrete homology groups at scale} \(r\). They prove that, for every \(r\), this sequence of groups defines a discrete homology theory in the usual sense, i.e. satisfying the Eilenberg-Steenrod axioms. It also satisfies the Mayer-Vietoris exact sequence. Extending the notion of discrete fundamental group of a graph given by \(A\)-theory, \(A_{1,r}(X,p)\), a \textsl{discrete fundamental group at scale} \(r\), is defined for every metric space \((X,d)\) with base point \(p\); it is proved that there is a Hurewicz map in dimension 1 NEWLINE\[NEWLINEA_{1,r}(X,p) \longrightarrow DH_{1,r}(X)NEWLINE\]NEWLINE showing that \(DH_{1,r}(X)\) is the abelianization of \(A_{1,r}(X,p)\).NEWLINENEWLINEIn the last section of the paper, by considering direct systems of homology groups at scale \(r\) with \(r\) going to infinity, the authors develop a coarse approach of discrete homology theory which studies metric spaces from a large-scale point of view and which was one of the main motivations for having worked in the general frame of metric spaces.