Torelli Actions and Smooth Structures on 4-manifolds

In concept of Artin presentations, a smooth 4 manifold is determined by an Artin presentation of the fundamental group of its boundary. Hence, among the main challenges in 4 dimensional smooth topology, specifically the research into smooth structures on these manifolds and their Donaldson and Seiberg-Witten invariants, could be approached in a totally fresh, external, purely group theoretic way. The primary reason for this dissertation is to clearly show the way to customize the smooth structure in this fashion. These particular instances also have physical relevance. We also solve some related problems. Specifically, we review knot and link theory in Artin presentation theory, provide a group theoretic formula for the Casson invariant, study the combinatorial group theory of Artin presentations, while stating a few crucial open challenges…

Contents: Torelli Actions and Smooth Structures on 4-manifolds

Chapter 1. Introduction
Chapter 2. Artin Presentations
2.1. Homeomorphisms of the punctured 2-disk
2.2. Pure braids
2.3. The 3-manifolds M3(r)
2.4. The 4-manifolds W4 (r)
Chapter 3. Torelli Actions
3.1. Proof of Theorem5
3.2. Identifying Boundaries
Chapter 4. Knots and Links
Chapter 5. The Casson Invariant in AP Theory
Chapter 6. Combinatorial Group Theory
iv6.1. Basic Properties of Free Groups
6.2. Proof of Theorem 10
6.3. ProofofTheorem9
6.4. Characterizing the ri..

Source: University of Maryland

Leave a Comment