论文标题
对广义的Toral Rank猜想的反例
Counter-Examples to a generalised Toral Rank Conjecture
论文作者
论文摘要
Toral等级的猜想推测，紧凑型歧管的Betti数字的总和承认，$ r $的圆环的自由动作从下面限制为$ 2^r $。显然，这样的动作产生了圆环束，更普遍地，对合适的拓扑圆环纤维的总空间进行了相同的共同结合，félix-oprea-oprea--tanré。 在本文中，我们表明，这种广义的Toral等级猜想无法通过为其提供各种不同的反例来实现（对于每个等级$ r \ geq 5 $）。特别是，我们表明，有一系列光滑的尼尔耐纤维束束，具有纤维的nilmanifolds，均具有等级$ r $的圆环，以使总空间和纤维的总维度的总维度和纤维的总尺寸收敛到$ 0 $，$ r $ $ r $ tondend to Infination to infinity to Infinity。 此外，我们证明，几乎没有免费的圆环动作无法实现我们的圆环纤维。更确切地说，在所描绘的序列中，束和圆环等级（恒定一个）在捆绑中的等级的差异甚至倾向于无穷大。 与沃克（Walker）最近的示例（完全不同的性质）类似，这表明，托拉尔（Toral）的猜想不太可能来自“较弱的猜想或结构”。
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 Félix--Oprea--Tanré. 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".
