Homotopy theory studies deformations of spaces and maps between them, an inherently continuous concept. In this talk, we will explore how this translates to the discrete category of graphs. We will see how to define homotopy for graphs, and show that we can break down a deformation of graph maps into moves of one vertex at a time. We also discuss how to recognize when two graphs are the same up to deformation (‘homotopy equivalent’), and look at developing homotopy invariants for graphs. No prior knowledge of either homotopy theory or graph theory will be needed, and many examples and pictures will be given. Parts of this work are joint with Dr. Tien Chih at MSU Billings, and parts were developed in collaboration with Fort Lewis College undergraduate students Coleman Kane, Diego Novoa and Jonathon Thompson.
Homotopy theory for graphs
Mar. 07, 2019 2pm (Math 350)
Functional Analysis
Laura Scull (Lewis College)
X
The equivariant fundamental groupoid was originally defined by Tom Dieck for spaces with group actions, as a way to get an invariant that captured more of the structure of the fixed sets of the action than the traditional fundamental group. This construction was later reinterpreted as a Grothendieck category. I will give the definition and examples illustrating the structure of this category. Then I will talk about how the categorical interpretation leads to a potential generalization to groupoids, and give an outline of how such an argument might go. This is work in progress, so we have not yet filled in all the steps necessary to get a well-defined orbifold invariant. I will give the results we have and explain what still needs to be done. This is joint work with Dorette Pronk at Dalhousie university.
An equivariant fundamental group for orbifolds Sponsored by the Meyer Fund