Jens Harlander (Boise State University)

Title: Geometry and topology of labeled oriented trees.

Abstract: Whitehead's asphericity conjecture states that a subcomplex of an aspherical 2-complex is aspherical. He formulated it as a question in 1941. Whitehead noticed that an affirmative answer implies the asphericity of knot complements, a fact unknown at the time. Papakyriakopoulos established the asphericity of knot complements in 1957 using 3-manifold topology and not relying on Whitehead's ideas. The Whitehead conjecture remains unresolved to this day. Generalized knot complements and their 2-dimensional spines remain at the center of the conjecture. Presentations for such spines can be encoded by labeled oriented trees, lot's for short. In my talk I will give a survey of the algebra, topology, and geometry surrounding labeled oriented trees.