Date & Time: Wednesday, February 17, 2016, 09:30-10:30
Venue: Ramanujan Hall
Title: Wild Topology, Group Theory and a conjecture in Number Theory
Speaker: Gregory Conner, Brigham Young University, Provo, USA
Abstract: One of the most useful analogies in mathematics is the fundamental group functor (also known as the Galois Correspondence) which sends a topological space to its fundamental group while at the same time sending continuous maps between spaces to corresponding homomorphisms of groups in such a way that compositions of maps are preserved. A an obvious question one might ask is whether the fundamental group functor is "onto", that is: (1) is every group the fundamental group of a space and (2) every homomorphism the image of a continuous map between corresponding spaces? The easy answer to (1) is "yes" and the nonobvious answer to (2) is "it depends on the spaces". We'll introduce the harmonic archipelago as the shining example of a space with a strange fundamental group, define an archipelago of groups as a group theoretic product operation and finally describe how such products are (almost) all isomorphic to the fundamental group of the harmonic archipelago. We will study examples showing that there are group homomorphisms that cannot be induced by continuous maps on certain spaces and how the fundamental group of the harmonic archipelago factors through all such "discontinuous homomorphisms", how none of the examples is constructible (or even understandable in any reasonable way) and how one might detect spaces whose fundamental group allows them to be the codomain of such weird homomorphisms (the conjecture is that they contain the rational numbers or torsion). We'll talk a bit about the notion of cotorsion groups from classical Abelian group theory and how that notion can be generalized to non-Abelian groups by requiring certain types of systems of equations have solutions and then mention how countable groups which have solutions to such systems are always images of the fundamental group of an archipelago. In the end we're lead to an example of a countable group which we can prove is either the rational numbers or gives a counterexample to a nearly 50 year old conjecture in number theory: the Kurepa conjecture. So there is a little topology, a little homotopy theory, some group theory, a pinch of logic and a wisp of number theory in the talk. This is a distillation of work I've published recently with Hojka and Meilstrup (Proc AMS) and work that is still being written up with Hojka and Herfort.