On macroscopic dimension of rationally inessential manifolds. (English. Russian original) Zbl 1271.53043

Dranishnikov, A. N.

Funct. Anal. Appl. 45, No. 3, 187-191 (2011); translation from Funkts. Anal. Prilozh. 45, No. 3, 34-40 (2011).

Summary: We show that, for a rationally inessential orientable closed \(n\)-manifold \(M\) whose fundamental group is a duality group, the macroscopic dimension of its universal cover \(\widetilde M\) is strictly less than \(n:\dim_{MC}\widetilde M<n\). As a corollary, we obtain the following partial result towards Gromov’s conjecture:
The inequality \(\dim_{MC}\widetilde M<n\) holds for the universal cover \(\widetilde M\) of a closed spin \(n\)-manifold \(M\) with a positive scalar curvature metric if the fundamental group \(\pi_1(M)\) is a duality group satisfying the analytic Novikov conjecture.

MSC:

53C23

Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces

Keywords:

macroscopic dimension; inessential manifold; duality group

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