Speaker: Alex Margolis
Title:Coarse homological invariants of metric spaces
Abstract: A classical theorem of Hopf and Freudenthal states that if G is a finitely generated group, then the number of ends of G is either 0, 1, 2 or infinity. We prove a higher-dimensional analogue of this result, showing that if F is a field, G is countable, and Hk(G,FG)=0 for k <n, then dim Hn(G,FG)=0,1 or ∞, significantly extending work of Farrell from 1975. Moreover, in the case dim Hn(G,FG)=1, then G must be a coarse Poincaré duality group. We prove an analogous result for metric spaces. In this talk, we talk about the tools needed to prove this result. We will introduce several coarse topological invariants of metric spaces, inspired by group cohomology. We define the coarse cohomological dimension of a metric space, and demonstrate that if G is a countable group equipped with a proper left-invariant metric, then the coarse cohomological dimension of G coincides with its cohomological dimension whenever the latter is finite. Extending a result of Sauer, we show that coarse cohomological dimension is invariant under coarse equivalence. We characterise unbounded quasi-trees as quasi-geodesic metric spaces of coarse cohomological dimension one.