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Abstract
In this thesis, we develop a topological framework for studying the twisted homology of certain families of braid subgroups, which arise naturally as the fundamental groups of various configuration spaces. Our construction relies on a new cellular stratification of configuration spaces of the plane with punctures, based on the classical Fox-Neuwirth stratification of configuration spaces of the plane. Using this new tool, we identify the homology of braid groups on punctured genus-0 surfaces with exponential coefficients arising from braided vector spaces, with the cohomology of certain bimodules over a quantum shuffle algebra. This structural theorem has several consequences. First, we give a complete characterization of the homology of Artin groups of type B with one-dimensional twisted coefficients over a field of characteristic 0. Second, we compute the homology of genus-0 surface braid groups and prove a vanishing range for the homology of mixed braid groups with certain one-dimensional twisted coefficients. Third, we prove an upper bound on the Betti numbers of Hurwitz spaces over punctured curves of genus 0. Our topological findings have applications in number theory. We give an upper bound on character sums of the resultant over pairs of monic squarefree polynomials of given degrees, answering and generalizing a question of Ellenberg and Shusterman. Finally, we sketch a blueprint for proving the upper bound in a version of the weak Malle's conjecture on the enumeration of finite extensions of Fq(t) with specified Galois group, bounded discriminant, and prescribed ramification at finitely many primes, refining a result of Ellenberg-Tran-Westerland.
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University of Minnesota Ph.D. dissertation. June 2024. Major: Mathematics. Advisor: Craig Westerland. 1 computer file (PDF); vii, 132 pages.
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Hoang, Anh. (2024). untitled - dbId: ac85f401-6e29-4458-a73e-2074bb89710b. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/276769.
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