## 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.

#### Recommended Citation

Orejola, Oliver A., "Cohomologous 2-cocycles are Homotopic 2-cocycles: k-graphs and C*-algebras" (2016). *Undergraduate Honors Theses*. 1076.

https://scholar.colorado.edu/honr_theses/1076