Destroying Tricommutativity of a 3-Cube Using the Symmetric Functor
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In the following thesis, in pursuit of a Master’s degree in Mathematics at Texas A &M International University (TAMIU), Laredo, Texas, we provide a modicum of background and introductory material to motivate the search for, and demonstrate examples of a satisfactory example of a tricommutative 3-cube whose tricommutativity can be “destroyed” through the employ of the symmetric square functor SP2. The successful identification of a counter example of such a 3-cube is done using projection maps on inverse limit systems and through the consideration of commutative diagrams of symmetric powers of spaces of weight 2ω2 . A previous counterexample of destroyed tricommutativity involving the Vietoris hyperspace operator, also known as the exponential functor, was previously demonstrated through the work of Mr. R. Montemayor, and this work extends the result in a topological context to the case of the symmetric functor. The existence of both such cases was proven by Dr. D. K. Milovich in other previous work. Much of the background draws from the previous literature of the advisor to the author, Dr. D. K. Milovich, the text and papers of L. Heindorf, L. Shapiro.