Title page for ETD etd-11042009-151333

Type of Document Dissertation
Author Egedy, Charles Richard
Author's Email Address cegedy1@tigers.lsu.edu
URN etd-11042009-151333
Title The Extended Picture Group, With Applications to Line Arrangement Complements
Degree Doctor of Philosophy (Ph.D.)
Department Mathematics
Advisory Committee
Advisor Name Title
Cohen, Daniel Committee Chair
Hofmann, Jerome Committee Member
Oporowski, Bogdan Committee Member
Sengupta, Ambar Committee Member
Smolinsky, Lawrence Committee Member
Busch, Konstantin Dean's Representative
  • algebraic topology
  • higher homotopy
Date of Defense 2009-10-23
Availability unrestricted
We obtain the picture group as the quotient with a torsion subgroup, of an extended picture group, which is isomorphic to the kernel of a precrossed module homomorphism. In addition to expanding the notion of a picture group, the new formulation gives a natural way to construct homomorphisms between picture groups by describing deformations of one-vertex subpictures. The extended picture group thus provides a convenient way to describe generators for the second homotopy group of line arrangement complements as well as homomorphisms between these groups. In particular, we show that the homomorphisms relate to a lattice structure corresponding roughly to the condition of being more nearly in general position. Examples include generators for Falk's X2 arrangement and for a generic section of braid arrangement A3. Finally, we demonstrate that the C3 arrangement C(5) is a K(pi; 1) space.
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