Torelli Actions and Smooth Structures on 4-manifolds

Abstract

In the theory of Artin presentations, a smooth four manifold is already determined by an Artin presentation of the fundamental group of its boundary. Thus, one of the central problems in four dimensional smooth topology, namely the study of smooth structures on these manifolds and their Donaldson and Seiberg-Witten invariants, can be approached in an entirely new, exterior, purely group theoretic manner. The main purpose of this thesis is to explicitly demonstrate how to change the smooth structure in this manner. These examples also have physical relevance. We also solve some related problems. Namely, we study knot and link theory in Artin presentation theory, give a group theoretic formula for the Casson invariant, study the combinatorial group theory of Artin presentations, and state some important open problems.