| Frank Quinn has previously developed multiplicative invariants of 2 dimensional CW-complexes (2-complexes) by using concepts from Topological Quantum Field Theory. One motivation for this work 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 special 2-complex. Second, we introduce a category of algebraic objects, called double semigroups, 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.; We also give some examples to show how topological invariants can be useful in Systems Science. These invariants can give qualitative information about complex dynamical systems and emergent structures. |
