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.
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:
hx_{1},...,x_{n }| r_{1},...,r_{n}i
satisfying the following equation in F_{n }(the free group on x_{1},...,x_{n}):
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 P_{n }× Z^{n}, where P_{n }is the classical pure
1
braid group on n strands. Let Ω_{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 Ω_{n }that is the identity on ∂Ω_{n }and is unique up to isotopy rel ∂Ω_{n}, |
M^{3 }(r) |
= |
a closed, connected, orientable 3-manifold, |
W^{4 }(r) |
= |
a smooth, compact, connected, simply connected 4-manifold. |