Kielak Dawid, Sun Bin
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG UK.
Present Address: 619 Red Cedar Road C212, Wells Hall, East Lansing, MI 48824 USA.
Math Ann. 2024;390(3):3567-3619. doi: 10.1007/s00208-024-02835-7. Epub 2024 Mar 26.
We prove that twisted -Betti numbers of locally indicable groups are equal to the usual -Betti numbers rescaled by the dimension of the twisting representation; this answers a question of Lück for this class of groups. It also leads to two formulae: given a fibration with base space having locally indicable fundamental group, and with a simply-connected fiber , the first formula bounds -Betti numbers of in terms of -Betti numbers of and usual Betti numbers of ; the second formula computes exactly in terms of the same data, provided that is a high-dimensional sphere. We also present an inequality between twisted Alexander and Thurston norms for free-by-cyclic groups and 3-manifolds. The technical tools we use come from the theory of generalised agrarian invariants, whose study we initiate in this paper.
我们证明，局部可指示群的扭曲(\mathcal{L}^2)-贝蒂数等于通过扭曲表示的维数重新缩放后的通常(\mathcal{L}^2)-贝蒂数；这回答了吕克针对此类群提出的一个问题。它还引出了两个公式：给定一个纤维化(p:E\to B)，其底空间(B)具有局部可指示的基本群，且纤维(F)是单连通的，第一个公式根据(B)的(\mathcal{L}^2)-贝蒂数和(F)的通常贝蒂数来界定(E)的(\mathcal{L}^2)-贝蒂数；第二个公式在(F)是高维球面的情况下，根据相同的数据精确计算(E)的(\mathcal{L}^2)-贝蒂数。我们还给出了自由循环群和三维流形的扭曲亚历山大范数与瑟斯顿范数之间的一个不等式。我们使用的技术工具来自广义耕地不变量理论，本文将开启对此理论的研究。
