Transfinite extension of asymptotic dimension
by
Taras Radul
Ivan Franko National University of Lviv

Asymptotic dimension asdim of a metric space was defined by M.Gromov for studying asymptotic invariants of discrete groups, particularly fundamental groups of manifolds [1]. Then G.Yu found a successful application of the asymptotic dimension to prove a series of conjectures, including the famous Novikov Higher Signature conjecture, for groups with finite asdim [2]. Many authors started to develop a general theory called asymptotic topology or large-scale geometry (see for example the survey [3]). Asymptotic topology studies global properties of (unbounded) metric spaces neglecting small (bounded) details of these spaces. Thus, the properties and invariants of metric spaces under consideration are treated in the limit, at infinity. The large-scale world, the macrocosm, admits an analogue with the microcosm, where the limit is taken at a point. A significant part of topology is devoted to the investigation of local properties of spaces and can thus be referred to the microcosm. Dimension theory can serve as example. It became important to carry over topological theories to macrocosm.

The dimension asdim can be considered as asymptotic analogue of the Lebesgue covering dimension dim. Dranishnikov has introduced dimensions asInd and asind which are analogous to large inductive dimension Ind and small inductive dimension ind [4, 5]. There many analogues and differences between topological and asymptotic dimension theories. We discuss basic theorems of dimension theory like sum theorem, addition theorem and subspace theorem.

M.Zarichnyi has proposed to consider transfinite extension of asInd and conjectured that this extension is trivial. This conjecture is proved in [6]: if a space has a transfinite asymptotic dimension, then its dimension is finite. The same result we obtain for asind.

We define as well a transfinite extension trasdim and show that this extension is not trivial: there exist metric spaces with transfinite infinite dimension.

[1] M.Gromov. Asymptotic invariants of infinite groups. Geometric group theory. v.2, Cambridge University Press, 1993.

[2] G.Yu. The Novikov conjecture for groups with finite asymptotic dimension // Ann. Math 147 (1998), 325-355.

[3] A.N. Dranishnikov. Asymptotic topology // Russian Math. Surveys. 55 (2000), 1085-1129.

[4] A.N. Dranishnikov. On asymptotic inductive dimension // IP J. Geom.Topol. 3 (2001), 239-247.

[5] A.N. Dranishnikov, M.M.Zarichnyi. Universal spaces for asymptotic dimension // Topology and Appl. 140 (2004), 203-225.

[6] T.Radul. Addition and subspace theorems for asymptotic large inductive dimension // Colloq. Math. 106 (2006), 57-67.

Date received: May 1, 2008