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Abstract:
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This dissertation examines simplicial complexes associated with cyclotomic po-lynomials and irreducible characters of finite solvable groups. In the process of analysis ofthe associated objects special attention is paid to the noncommutativity of the examinedstructures.A collection of simplicial complexes can be associated to an algebraic object such as acyclotomic polynomial. In most cases, the homotopy type of associated simplicial complexesgives us complete information about the coefficients of the cyclotomic polynomial. The onlyexceptions are cyclotomic polynomials whose degree is a product of three different primenumbers and this case is the focus of research in this doctoral dissertation. When it ispossible, the homotopy type of a simplicial complex associated with the polynomialΦpqr(x),wherep,qandrare different prime numbers, is determined by using the discrete Morsetheory. However, in special cases, the simplicial complexes associated with the polynomialΦpqr(x)have a noncommutative fundamental group, thus providing a new noncommutativeinvariant of this type of polynomial. Complex presentations that appear as presentations ofthe fundamental groups of associated simplicial complexes are analyzed using Fox’s calculus.This thesis also focus on the study of simplicial complexes associated to a set of irreduciblecharacters of a finite solvable group. Two types of simplicial complexes are attached to aset of irreducible characters of a finite solvable group — character degree complex and primedivisor complex. The examination of the fundamental group of these types of simplicial com-plexes provides better understanding of the structure of the irreducible characters of finitesolvable groups. |