DOCUMENTA MATHEMATICA, Extra Volume ICM II (1998), 423-432

A. N. Dranishnikov

Title: Dimension Theory and Large Riemannian Manifolds

In this paper we discuss some recent applications of dimension theory to the Novikov and similar conjectures. We consider only geometrically finite groups i.e. groups $\Gamma$ that have a compact classifying space $B\Gamma$. It is still unknown whether all such groups admit a sphere at infinity \cite{B}. In late 80s old Alexandroff's problem on the coincidence between covering and cohomological dimensions was solved negatively \cite{Dr}. This brought to existence a locally nice homology sphere which is infinite dimensional. In the beginning of 90s S. Ferry conjectured that if such homology sphere can be presented as a sphere at infinity of some group $\Gamma$, then the Novikov conjecture is false for $\Gamma$. Here we discuss the development of this idea. We outline a reduction of the Novikov conjecture to dimension theoretic problems. The pionering work in this direction was done by G. Yu [Yu]. He found a reduction of the Novikov Conjecture to the problem of finite asymptotic dimensionality of the fundamental group $\Gamma$. Our approach is based on the hypothetical equivalence between asymptotical dimension of a group and the covering dimension of its Higson corona. The slogan here is that most of the asymptotic properties of $\Gamma$ can be expressed in terms of topological properties of the Higson corona $\nu\Gamma$. At the end of the paper we compare existing reductions of the Novikov conjecture in terms of the Higson corona.

1991 Mathematics Subject Classification:

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