PTM-DMV Conference: Wild algebraic & geometric topology session

I realize it has been a while since I have posted anything. I’m afraid teaching and research took up enough of my late summer schedule to keep me from finishing up a number of posts I had started. This is alright; I’ve been working hard trying to make my first experience with teaching topology at Georgia State a success.

This post is a reminder about the upcoming PTM-DMV Conference in Poznan, Poland: the joint meeting of the German Mathematical Society (Deutsche Mathematiker-Vereinigung) and Polish Mathematical Societies (Polskie Towarzystwo Matematyczne).

I will be giving a talk in the Session on Wild Algebraic & Geometric Topology.

Abstracts can be found on the session link above. Here is the schedule:

9/17/2014 (Wednesday)

2:30 – 2:55 PM Wolfgang Herfort – Cotorsion and Homology
3:00 – 3:25 PM Wolfram Hojka – Mapping the harmonic archipelago
3:30 – 3:55 PM Isacc Goldbring – The fundamental group of a locally finite graph with ends: a hyperfinite approach
4:00 – 4:25 PM Benoit Loridant – Fundamental group of Rauzy fractals
4:30 – 5:00 PM Coffee Break
5:00 – 5:25 PM Jean-Francois LaFont – One-dimensional geodesic spaces, Part I: Structure Theory
5:30 – 5:55 PM David Constantine – One-dimensional geodesic spaces, Part II: Marked length rigidity

9/18/2014 (Thursday)

2:30 – 2:55 PM Hanspeter Fischer – Word calculus in the fundamental group of the Menger Curve
3:00 – 3:25 PM Katsuya Eda – Singular homology groups of one-dimensional Peano continua
3:30 – 3:55 PM Janusz Przewocki – Milnor-Thurston homology of some wild topological spaces
4:00 – 4:25 PM Thilo Kuessner – Measure homology and singular homology
4:30 – 5:00 PM Coffee Break
5:00 – 5:25 PM Jeremy Brazas – A characterization of the unique path lifting property for the whisker topology
5:30 – 5:55 PM Ali Pakdaman – One point unions preserve having the categorical universal covering
6:00 – 6:25 PM Alvaro Sanchez-Gonzalez – A shape topology for the universal path space

9/19/2014 (Friday)

2:30 – 2:55 PM Matija Cencelj – Gropes and their fundamental groups
3:00 – 3:25 PM Brendon LaBuz – Big free groups acting on $\Lambda$ -trees
3:30 – 3:55 PM Oleg Bogopolski – Generalized presentations of groups, in particular of $Aut(F_{\omega})$
4:00 – 4:25 PM Andreas Zastrow – The obstruction to contractibility of snake cones and Alternating cones
4:30 – 5:00 PM Coffee Break

There should be many great talks in this session on some pretty wild stuff!

My own talk will address the following theoretical barrier: to what extent can the structure of the fundamental group of a path-connected metric space be understood using generalized covering maps based on unique lifting of paths and homotopies of paths? Various generalizations have proven to be particularly useful for studying one-dimensional spaces like the Menger curve (and will be used in Hanspeter Fischer’s talk) and studying free topological groups.

I’ll use some categorical ideas to show that if a subgroup $H\subset\pi_1(X)$ corresponds to any such covering-like map (with the most primitive unique lifting properties), then it can be understood using generalized coverings constructed with the so-called “whisker topology.” The whisker construction $\widetilde{X}_H$ is not specialized, it appears in most textbooks that include the classification of covering spaces, however, it doesn’t not seem to be as well-known that it has much more general applications. The obstruction to whether or not these generalized coverings exists is precisely the unique path lifting property of the endpoint projection $p_H:\widetilde{X}_H\to X$ . I’ll then characterize, for any arbitrary subgroup $H$ , the existence of unique path lifting for this map in terms of a more practical sequential closure-like condition using test maps from the following one-dimensional planar Peano continuum $\mathbb{D}$ .