Homological Growth of Groups and Aspherical Manifolds

拓扑学通常被描述为在橡皮板上做几何；如果一个空间可以在不切割或粘合的情况下转换成另一个空间，则被认为是等效的。同调是一个基本的代数不变量，可以区分到这个等价的空间。例如，甜甜圈表面的同源性与甜甜圈孔表面的同源性不同，这意味着将其中一个表面变成另一个表面需要一些切割或粘合。同调不变量通常很难计算，特别是在高维中。这个项目不是精确地计算它们，而是解决关于自然发生的空间序列的这些不变量的增长率的问题。提出的研究集中在非球面流形拓扑的几个长期猜想。该项目还将通过指导和外展促进研究生和本科生教育。更具体地说，本研究计划的主要目标是研究非球面复合体有限正则复盖残塔中同调不变量的增长。潜在的动机是一个猜想，即在这样的塔中有理贝蒂数的线性增长应该阻止这些复合体同伦等价于一定维数的流形。本课题将用模p或积分同调代替有理同调来探讨这个猜想及其变体。例如，PI计划构造Gromov双曲群，其中Betti数的增长率取决于系数域，以及局部CAT(-1)，具有mod p个Betti数线性增长的奇维流形。这样的流形实际上不会在一个圆上纤维化，PI将进一步探索消失的同调增长和Bieri-Neumann-Strebel不变量（编码这种纤维化的代数类似物）之间的联系。PI还将研究非球面复合体的流形增厚，其中这种同构生长消失，以及粗糙几何产生的其他障碍。该项目由拓扑学和促进竞争研究的既定计划（EPSCoR）共同资助。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Topology is often described as doing geometry on a rubber sheet; spaces are considered equivalent if one can be transformed into the other without cutting or gluing. Homology is a fundamental algebraic invariant which can distinguish spaces up to this equivalence. For instance, the homology of the surface of a donut is different than that of the surface of a donut hole, and this implies that turning one of these surfaces into the other requires some cutting or gluing. Homological invariants are generally difficult to compute, especially in higher dimensions. Instead of computing them exactly, this project addresses questions about the growth rate of these invariants for naturally occurring sequences of spaces. The proposed research concentrates on several long-standing conjectures on the topology of aspherical manifolds. This project will also promote graduate and undergraduate education through mentoring and outreach. More specifically, the primary goal of the research program is to study the growth of homological invariants in a residual tower of finite regular covers of an aspherical complex. The underlying motivation is a conjecture that linear growth of rational Betti numbers in such a tower should obstruct these complexes being homotopy equivalent to manifolds of a certain dimension. The project will explore this conjecture and variants of it with rational homology replaced with mod p or integral homology. For instance, the PI plans to construct Gromov hyperbolic groups where the growth rate of Betti numbers depends on the field of coefficients, and locally CAT(-1), odd-dimensional manifolds with linear growth of mod p Betti numbers. Such manifolds will not virtually fiber over a circle, and the PI will further explore the connection between vanishing homological growth and the Bieri-Neumann-Strebel invariants (which encode algebraic analogues of such fibering). The PI will also study manifold thickenings of aspherical complexes where this homological growth vanishes and other obstructions from coarse geometry arise. This project is jointly funded by Topology and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.