Jack S. Calcut Research Statement 1.2 Quotient Maps and the Fundamental Group 1.3 the Pullback Functor on Coverings

The concept of a topology for the fundamental group appears to have originated with Hurewicz (1935), but it has been actively studied only for the last ten years. The classical fundamental group of a pointed topological space X naturally inherits the quotient topology from the space of based loops in X endowed with the compact-open topology. The resulting space, called the topological fundamental group, can contain more information than the fundamental group, as shown by Fabel [F1]. An immediate question is to understand when the topological fundamental group is discrete. John McCarthy and I proved (2009) a folklore theorem stating that if X is a locally path connected topological space, then the topological fundamental group of X is discrete if and only X is semilocally simply connected [CM]. Our work was recently cited in the preprints [B] and [F2] both of which show that multiplication is not continuous in general in the topological fundamental group.