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.

Committee Chair

Cohen, Daniel

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