ABSTRACT This dissertation constructs total invariants to distinguish between simple homotopy classes of simply connected 2-complexes. Frank Quinn developed multiplicative invariants of 2 dimensional CW-complexes (2-complexes) by using concepts from Topological Quantum Field Theory. One motivation for this dissertation was to get invariants that could survive stabilization and therefore potentially distinguish Andrews-Curtis classes of 2-complexes. Such invariants would help resolve the well known Andrews Curtis Conjecture that no nontrivial Andrews Curtis Classes exist. In this dissertation we follow Quinn’s philosophy, but from the top down. We first construct a canonical decomposition of a special 2-complex. Second, we introduce a category of algebraic objects, called a double semigroup, which allows free constructions and quotients. Next, we build a particular double semigroup whose elements are a complete set of invariants for homeomorphism classes of special 2-complexes. Finally, we construct a double semigroup whose elements are a complete set of Andrews Curtis invariants. Tuesday, June 11, 2002 DISSERTATION COMMITTEE M. Paul Latiolais, Chairman Andrew M. Fraser Joyce O’Halloran Serge Preston Erik Bodegom, Graduate Studies Rep.

root 2004-05-05