Rational manifold models for duality groups

TL;DR

This paper demonstrates that certain duality groups can be realized as fundamental groups of high-dimensional manifolds with rationally acyclic universal covers, leading to new examples that challenge existing conjectures about $L^2$-Betti numbers.

Contribution

It establishes a link between duality groups and high-dimensional manifolds with rationally acyclic universal covers, providing counterexamples to a rational version of Singer's conjecture.

Findings

Construction of manifolds with rationally acyclic universal covers

Existence of nonvanishing $L^2$-Betti numbers outside the middle dimension

Counterexamples to a rational analogue of Singer's conjecture

Abstract

We show that a finite type duality group of dimension $d>2$ is the fundamental group of a $(d+3)$-manifold with rationally acyclic universal cover. We use this to find closed manifolds with rationally acyclic universal cover and some nonvanishing $L^2$-Betti numbers outside the middle dimension, which contradicts a rational analogue of a conjecture of Singer.