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Dissertation
The Milnor fiber associated to an arrangement of hyperplanes
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Summer 2011
DOI: 10.17077/etd.wewwtrs3
Abstract
<p>Let <em>f</em> be a non-constant, homogeneous, complex polynomial in <em>n</em> variables. We may associate to <em>f</em> a fibration with typical fiber <em>F</em> known as the Milnor fiber. For regular and isolated singular points of <em>f</em> at the origin, the topology of the Milnor fiber is well-understood. However, much less is known about the topology in the case of non-isolated singular points. In this thesis we analyze the Milnor fiber associated to a hyperplane arrangement, ie, <em>f</em> is a reduced, homogeneous polynomial with degree one irreducible components in <em>n</em> variables. If <em>n</em> > 2then the origin will be a non-isolated singular point. In particular, we use the fundamental group of the complement of the arrangement in order to construct a regular CW-complex that is homotopy equivalent to the Milnor fiber. Combining this construction with some local combinatorics of the arrangement, we generalize some known results on the upper bounds for the first betti number of the Milnor fiber. For several classes of arrangements we show that the first homology group of the Milnor fiber is torsion free. In the final section, we use methods that depend on the embedding of the arrangement in the complex projective plane (ie not necessarily combinatorial data) in order to analyze arrangements to which the known results on arrangements do not apply.</p>
Details
- Title: Subtitle
- The Milnor fiber associated to an arrangement of hyperplanes
- Creators
- Kristopher John Williams - University of Iowa
- Contributors
- Richard Randell (Advisor)Charles Frohman (Committee Member)Maggy Tomova (Committee Member)Paul Muhly (Committee Member)Julianna Tymoczko (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Summer 2011
- Publisher
- University of Iowa
- DOI
- 10.17077/etd.wewwtrs3
- Number of pages
- vi, 79 pages
- Copyright
- Copyright 2011 Kristopher John Williams
- Language
- English
- Description bibliographic
- Includes bibliographical references (pages 78-79).
- Academic Unit
- Mathematics
- Record Identifier
- 9983776633102771
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