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Macroscopic dimension and essential manifolds. (English) Zbl 1230.54027

Definition. A map \(f : X \to K\) between metric spaces \(X, K\) is called uniformly cobounded if there is \(D > 0\) such that \(diam(f^{-1} (y)) \leq D\).
A metric space \(X\) has the macroscopic dimension \(\dim_{MC} X \leq n\) if there is a uniformly cobounded proper continuous map \(f : X \to K^n\) to an \(n\)-dimensional polyhedron \(K^n\).
The Gromov Conjecture in this paper means: For an \(n\)-dimensional manifold \(M\) with positive scalar curvature, its universal covering is at most \((n-2)\)-dimensional from the macroscopic point of view.
The author proves this conjecture for all rational essential manifolds with amenable fundamental groups. For the general case he defines an obstruction for \(n\)-dimensional \(n\)-manifolds to the inequality \(\dim_{MC} \tilde{M} < n\). The obstruction is an element of the \(n\)-dimensional cohomology group of the universal cover \(\tilde{M}\) defined by means of “almost equivariant cochains”.

MSC:

54F45 Dimension theory in general topology
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