Laura Scull, Fort Lewis College
 
X-homotopy for graphs and Z/2-equivariance

Tuesday, Nov 15th, 12:pm

Abstract: X-homotopy for graphs is defined using the categorical product of graphs. In this context, our graphs are rigid and edges cannot be collapsed. Thus there is a sense in which graphs in this theory are built out of edges, instead of vertices. The 'flip' which reverses direction of an edge introduces a Z/2-action in this wordview, and X-homotopy phenomena such as parity can be then understood as Z/2-equviariant features.
In this talk, I will focus on the X-fundamental group. I will introduce this fundamental group in a classical way, as homotopy classes of paths in our graph, and give examples (and pictures). Then I will show how taking an edge-centric approach allows us to define a fundamental group as a Z/2-equivariant fundamental group of paths of edges. I will motivate the equivariant theory and then explain this second definition. These two seemingly different definitions give equivalent groups, providing a way of seeing the parity showing up in the traditional definition as coming from a Z/2-action.