Dots and Lines: A Combinatorial Interpretation of the Homotopy Groups of Finite Topologies

Overview

Objectives/Goals The homotopy groups of a topological space are commonly studied as a topological invariant which gives information about the space's holes, homotopy type, and connectedness. McCord showed the the homotopy groups are precisely those of corresponding simplicial complexes, but gives no explicit way of constructing or interpreting elements of the homotopy groups of a finite space, making applications difficult. An attempt to construct an analogue of homotopy groups for graphs was examined by Atkin and Smith, but the groups they associate to a graph G do not in general correspond to the homotopy groups of a finite topology whose associated graph is G when such a topology exists. Methods/Materials My project presents a new definition of the homotopy groups of a graph G, in the spirit of Atkin by using only combinatorial methods, and shows that these homotopy groups of G are indeed the same as any topology whose associated graph is G. In particular, this implies that two topologies with the same associated graph have isomorphic homotopy groups. This new definition allows for a construction with no known analogue in topological homotopy theory, as a graph which reduces the dimension of each homotopy group. More explicitly, for a graph G, I construct a graph G^k whose nth homotopy groupis the same as the (n+k)th homotopy group of G, for all nonnegative integers n. This also allows for algorithms converging to presentations for any homotopy group of any graph (and any simplicial complex after using McCord's correspondence.) Conclusions/Discussion The question of the existence of solutions to a certain class of communications routing-type problems is also considered, and using homotopy groups I provide a method of phrasing such problems in terms of groups, which generally simplifies the solution. In fact, in certain cases the existence of a solution can be determined within O(v^3) time (where v is the number of vertices). These problems occur in a variety of situations including chip and network design.

Awards (1)

• First Place

Competition history

• CSEF 2013 Mathematics & Software · Entry S1401

Resources

• CSEF project summary ↗

Related projects

Closest projects by meaning, across every fair and year in the corpus.

Browse more like this

• All Computational Sciences projects
• CSEF Computational Sciences projects
• Every CSEF 2013 project
• CSEF First Place winners