TORELLI ACTIONS AND SMOOTH STRUCTURES ON 4-MANIFOLDS

Diploma

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, while preserving the underlying continuous topological structure. 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.

Introduction

Artin Presentation theory (AP theory) is a discrete, purely group theoretic theory of smooth, compact, simply connected 4-manifolds, their boundaries, and knots and links therein [W],[CW].

By definition, an Artin presentation r is a finite presentation:

hx1,...,xn | r1,...,rni

satisfying the following equation in Fn (the free group on x1,...,xn):

x1x2 ···xn = ¡r1−1x1r1¢¡r2−1x2r2¢···¡rn−1xnrn¢.

The name, given by González-Acuña in 1975, was well chosen as Emil Artin first considered such presentations in 1925, p.416-441, regarding his theory of braids.

Details and proofs of the following statements are in [W],[CW], and they appear in Chapter 2 for the sake of completeness.

For n > 0, Rn will denote the set of Artin presentations on n generators. Rn forms a group canonically isomorphic to Pn × Zn, where Pn is the classical pure

1

braid group on n strands. Let n denote the compact 2-disk with n holes. An Artin presentation r ∈ Rn determines:

π (r)

=

the group presented by r,

A(r)

=

an n × n symmetric integer matrix,

h(r)

=

a self homeomorphism of n that is the identity

on n and is unique up to isotopy rel n,

M3 (r)

=

a closed, connected, orientable 3-manifold,

W4 (r)

=

a smooth, compact, connected, simply connected 4-manifold.