The explicit algebraic autonomy of Artin presentation theory and the Fox Calculus. I

Abstract

Given an Artin presentation r, we use the Fox Calculus to obtain a short, purely algebraic computation, just in function of r, of the second homotopy group of the associated manifold \(M^3(r)\). This allows us to give a Langlands-like formulation (not yet a proof) of the three-dimensional Borel conjecture for closed, orientable 3-manifolds, the latter being a theorem of geometrization theory and implying the Poincare conjecture.