## 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

### Keywords:

Gromov conjecture; macroscopic dimension
Full Text:

### References:

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