Homology growth in topology and group theory

Louisiana State University, Baton Rouge LA

Investigators

Abstract

The mathematics of this research project is in the area of topology, which studies spaces up to continuous deformation. Here, spaces are considered the same if one can be transformed into the other without cutting or gluing. One is often interested in computing algebraic invariants which can distinguish spaces up to this equivalence. One such invariant is homology, which lets one study the shape of spaces using linear algebra. In low dimensions, homology measures very concrete aspects of a space; for instance it detects the number of loops in a graph or the number of holes in a surface. In high dimensions, it measures more complicated features, and is generally more difficult to compute. The PI will study several long-standing conjectures on the topology of aspherical manifolds, as well as recent breakthroughs connecting homology growth to various aspects of fibering. This project will also promote graduate and undergraduate education through the writing of a textbook on L^2-homology and the development of new Vertically Integrated Research courses at Louisiana State University. The primary goal of the research program is to study the growth of homology (with various coefficients) in a residual tower of finite regular covers of an aspherical manifold. One of the motivating conjectures is that for a large class of closed, aspherical manifolds, sublinear homology growth in all degrees and all field coefficients implies that the manifold has a finite cover which fibers over the circle. This is a key step towards developing a high-dimensional analogue of Agol's Virtual Fibering Theorem ​in dimension 3. In one direction, the PI plans to construct closed, aspherical Gromov hyperbolic manifolds of any odd dimension which have nontrivial F_p-homology growth. The PI will also study fibering (and various group theoretic analogues) for Gromov-Thurston manifolds and other classical examples. In a related direction, the PI will continue work on Singer conjecture's on vanishing of L^2-homology groups for certain locally CAT(0) cubulated manifolds. In the final part of the project, the PI will investigate Atiyah's conjecture on integrality of L^2-Betti numbers for some classes of Artin groups. 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.