Macroscopic dimension and duality groups

TL;DR

This paper proves that for certain manifolds with duality group fundamental groups, the macroscopic dimension of their universal covers is less than the manifold's dimension, especially under conditions like positive scalar curvature and the Analytic Novikov Conjecture.

Contribution

It establishes a new inequality relating macroscopic dimension and duality groups, extending to manifolds with positive scalar curvature under specific group-theoretic conditions.

Findings

Macroscopic dimension of universal covers is less than the manifold dimension for duality group fundamental groups.

The inequality holds for spin manifolds with positive scalar curvature if the fundamental group satisfies the Analytic Novikov Conjecture.

Results connect geometric properties of manifolds with algebraic properties of their fundamental groups.

Abstract

We show that for a rationally inessential orientable closed $n$-manifold $M$ whose fundamental group $\pi$ is a duality group the macroscopic dimension of its universal cover is strictly less than $n$:$$ \dim_{MC}\Wi M<n.$$ As a corollary we obtain the following 0.1 Theorem. The inequality $ \dim_{MC}\Wi M<n$ holds for the universal cover of a closed spin $n$-manifold $M$ with a positive scalar curvature metric if the fundamental group $\pi_1(M)$ is a virtual duality group virtually satisfying the Analytic Novikov Conjecture.