Georgia Institute of Technology College of Sciences

We say that a Riemannian manifold has good higher expansion if every rationally null-homologous i-cycle bounds an i+1 chain of comparatively small volume. The interactions between expansion, spectral geometry, and topology have long been studied in the settings of graphs and surfaces. In this talk, I will explain how to construct rational homology 3-spheres which are good higher expanders. On the other hand, I will show that such higher expanders must be rather topologically complicated; in particular, we will demonstrate a super-polynomial-in-volume lower bound on their torsion homology.