Counter-Examples to a generalised Toral Rank Conjecture

Abstract

The toral rank conjecture speculates that the sum of the Betti numbers of a compact manifold admitting a free action of a torus of rank $r$ is bounded from below by $2^r$. Clearly, such an action yields a torus bundle, and, more generally, the same cohomological bound is conjectured for total spaces of suitable topological torus fibrations by F\'elix--Oprea--Tanr\'e. In this article we show that this generalised toral rank conjecture cannot hold by providing various different counter-examples to it (for each rank $r\geq 5$). In particular, we show that there are sequences of smooth nilpotent fibre bundles of nilmanifolds with fibre a torus of rank $r$ such that the quotient of the total dimensions of the cohomologies of total space and fibre even converges to $0$ with $r$ tending to infinity. We moreover prove that none of our torus fibrations can be realised by almost free torus actions. More precisely, in the depicted sequence the difference of the ranks of the torus fibres in the bundle and the toral ranks (which are constant one) even tends to infinity. Similar to recent examples by Walker (of a completely different nature) this shows that the toral rank conjecture is not likely to follow from "weaker conjectures or structures".