Applications of finite spaces to the study of the asphericity of LOT complexes

CERDEIRO, MANUELA ANA

Medellín

Workshop; 5to Encuentro Colombiano de Combinatoria; 2016

CIMPA - Universidad Nacional de Colombia - Universidad de Antioquia

In this work we study the problem of the asphericity of LOT complexes (Labeled Oriented Tree). This class of 2-dimensional CW-complexes appears in the study of certain manifolds that arise as complements of ribbon discs, in a generalization of classic knot theory. An arcwise-connected topological space is said to be aspherical if its homotopy groups πn (X) are trivial for all n ≥ 2. In the case of a 2-dimensional complex X, the condition π2(X) = 0 implies that X is aspherical. The conjecture about the asphericity of knot complements was an openproblem for a long time, and it was finally proved by Papakyriakopoulos  using methods of 3-manifolds. It is now conjectured that, as in the case of knots, the ribbon disc complements are also aspherical. This problem is strongly related to other problems in algebraic topology, group theory and differential topology. Although some partial results have been achieved, very little is known at the moment about the asphericity of LOT complexes. In this work we approach the problem from a new point of view, taking it to the context of finite topological spaces, where we can apply new techniques that make use of the gemetric and combinatorial nature of finite spaces. Using recent results of Barmak and Minian aboutG-colorings of finite spaces, we obtain an efficient description of the second homotopy group of a LOT complex, as a submodule of a free module, where the generators are indexed by the edges of the LOT and the equations are indexed by its vertices. Using this description, we find a method for the analysis of the asphericity of these complexes. With this new method we obtain results about the asphericity of important families of LOTs. These results are part of my doctoral thesis.