Gromov’s macroscopic dimension conjecture

Abstract

The following definition was given by M Gromov [2]: Definition 1.1 Let V be a metric space. We say that dim V k if there is a k – dimensional polyhedron P and a proper uniformly cobounded map W V !P such that Diam. .p// for all p 2P . A metric space V has macroscopic dimmc V k if dim V k for some possibly large <1. If k is minimal, we say that dimmc V D k . Gromov also stated the following questions which, for convenience, we state in the form of conjectures: C1 Let .M ;g/ be a closed Riemannian n–manifold with torsion-free fundamental group, and let .  M ; z g/ be the universal covering of M n with the pullback metric. Suppose that dimmc.  M ; z g/ < n. Then dimmc.  M ; z g/ < n 1. In [1] we proved C1 for the case nD 3. Evidently, the following conjecture would imply C1 (see also (C) of Section 2): C2 Let M n be a closed n–manifold with torsion-free fundamental group and let f W M ! B be a classifying map to the classifying space B . Suppose that f is homotopic to a mapping into the .n 1/–skeleton of B . Then f is in fact homotopic to a mapping into the .n 2/–skeleton of B . In this note we show that both conjectures fail for n 4. We always assume that universal covering are equipped with the pullback metrics.