Gregory Lupton, Cleveland State University
 
A Second Homotopy Group in Digital Topology

Wednesday, Nov 15th, 11:00am

Abstract: (Joint work with Musin, Scoville, Staecker and Trevino)
Digital topology refers to the use of notions and methods from (algebraic) topology to study digital images. A digital image in our sense is an idealization of an actual binary digital image, which consists of pixels in the plane, or higher dimensional analogues of such. One aim of digital topology is to provide useful theoretical background for certain steps of image processing, such as feature extraction or recognition.
An extensive literature on digital topology includes several treatments of the fundamental group. I will report on our recent work in which we develop a notion of a second (higher) homotopy group and calculate its value to be Z for a digital image that may reasonably be interpreted as a digital 2-sphere. This calculation involves some interesting combinatorial ingredients. Our development and calculation may equally well be applied in the (graph-theoretic) settings of tolerance spaces and X-homotopy theory. The definitions readily extend to higher homotopy groups of any dimension.
I will also speculate briefly about possible applications of our work.