Undergraduate Honors Theses

Thesis Defended

Spring 2016

Document Type

Thesis

Type of Thesis

Departmental Honors

Department

Mathematics

First Advisor

Dr. Elizabeth Gillaspy

Second Advisor

Dr. Nathaniel Thiem

Third Advisor

Dr. Oliver DeWolfe

Abstract

k-graphs equipped with a defined 2-cocycle allow one to construct examples of C-algebras. For some k-graph, Λ, there does not necessarily exist a unique choice of 2-cocycle, such that for Λ there may exist multiple C-algebras depending on one’s choice of 2-cocycle. There exist relations between two defined 2-cocycles on a k-graph, cohomology and homotopy, that imply features of the C∗-algebras generated by the 2-cocycles and Λ. Cohomology implies that the two C-algebras are isomorphic, and homotopy implies the two share the same invariant, K-theory. It is shown here the result that any two 2-cocycles defined on a k-graph which are cohomologous are then homotopic. Also included is the method by which one may construct a matrix equation,Ψx = z, that encodes the information of a k-graph and 2-cocycle, where the existence of an integer solution to the equation Ψx = z implies any two 2-cocycles are homotopic.

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