Identifier
etd-11042009-151333
Degree
Doctor of Philosophy (PhD)
Department
Mathematics
Document Type
Dissertation
Abstract
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.
Date
2009
Document Availability at the Time of Submission
Release the entire work immediately for access worldwide.
Recommended Citation
Egedy, Charles Richard, "The extended picture group, with applications to line arrangement complements" (2009). LSU Doctoral Dissertations. 2159.
https://repository.lsu.edu/gradschool_dissertations/2159
Committee Chair
Cohen, Daniel
DOI
10.31390/gradschool_dissertations.2159