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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