Multiplicative Invariants of Special 2-Complexes

Multiplicative Invariants of Special 2-Complexes [5]
John Rosson(Portland State)

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.