G. Conner: Some notes on discontinuous homomorphisms, W. Hojka: Mapping the harmonic archipelago

Gregory Conner, Some notes on discontinuous homomorphisms.

Abstract: One of the more useful tricks in topology and group theory is to move between the two using the fundamental group functor. It's a surprising fact that while every group is the fundamental group of a space, not every homomorphism of groups is induced by continuous maps of spaces. We will start with some background on fundamental groups of complicated spaces and then discuss various results related to such ``inscrutable'' homomorphisms. In particular, these can’t come up in everyday life. If time permits we’ll discuss how this seemingly technical topic relates to classical abelian group theory and has significant implications towards modern attempts at understanding fundamental groups of wild spaces.

 

 

Wolfram Hojka, Mapping the harmonic archipelago

Abstract: The study of wild algebraic topology has in the last decade seen an increased interest in two or higher dimensional spaces where nontrivial loops can be homotoped arbitrarily close to a point. The harmonic archipelago is a standard example with this property. The space is homeomorphic to a disc but for a single point and can be described as the reduced suspension of the graph of the topologist's sine curve y = sin(1/x).

The fundamental group of this space has peculiar mapping properties. For example, every countable locally free group embeds in G as a subgroup (hence so does the fundamental group of the complement of Alexander's horned cell!). In turn, every separable profinite group is an epimorphic image, as is every cotorsion group of at most continuum cardinality.