ARTIN PRESENTATIONS I: GAUGE THEORY, 3 + 1 TQFT'S AND THE BRAID GROUPS

Abstract

We inititate the systematic study of Artin Presentations, (discovered in 1975 by González-Acuña), which characterize the fundamental groups of closed, orientable 3-manifolds, and form a discrete equivalent of the theory of open book decompositions with planar pages of such manifolds. We list and prove the basic properties, state some fundamental problems and describe some of the advantages of the theory: e.g., an Artin Presentation of π1 (M3) does not just determine the closed, orientable 3-manifold M3, but also a canonical, smooth simply-connected cobordism of it, allowing us to tap into 4-dimensional gauge theory (and 3 + 1 TQFT's) in a more direct, purely discrete, functorial manner than others. Thus, in section 4, instead of using PDE's, we show how a canonical action of the commutator subgroup [Pn, Pn] of the pure braid group Pn can be used to study the smooth structures on a closed, smooth-connected 4-manifold with b2 = n, in a systematic way. However, the main purpose of this first paper is to Artin Presentations to set up simple criteria, testable with, say, MAGMA on the computer (where then no knowledge of topology is required) for finding explicit counter-examples to the so-called Weak Poincaré Conjecture: "Every homotopy 3-sphere bounds a smooth, compact, contractible 4-manifold," as well as: "Every irreducible Z-homology 3-sphere Σ, with π1 (Σ) = I (120) is homeomorphic to Σ (2, 3, 5)" and other conjectures implied by Thurston's Geometrization Conjecture. One first philosophical goal is to convince the reader that the truth of these conjectures is at least as unlikely as that of the Andrews-Curtis Conjecture and that ultimately, Artin Presentation Theory is a non-trivial intersection of string/M theory number theory.