Multiplicative Invariants of Special 2-Complexes
John Rosson
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
.