Intersection Form for 4-Manifolds, Knots, 2-Knots, Smooth Exotics & TGD

Abstract

The existence of exotic smooth structures even in the simplest possible 4-D space R⁴ might have some relevance for TGD. The study of the smooth structures in 4-D case involves intersection form for 2-homology of the 4-manifold. However, the existence of smooth structures in the 4-D case is not the only reason to get interested in this topic. The first reason is that in the TGD framework the intersection form describes the intersections of string world sheets and partonic 2-surfaces and therefore is of direct physical interest. The second reason relates to the role of knots in TGD. The 1-homology of the knot complement characterizes the knot. Time evolution defines a knot cobordism as a 2-surface consisting of knotted string world sheets and partonic 2-surfaces. A natural guess is that the 2-homology for the 4-D complement of this cobordism characterizes the knot cobordism. Also 2-knots are possible in 4-D space-time and a natural guess is that knot cobordism defines a 2-knot. Exotic smoothness could be anomalous in the TGD framework. Can one find any argument excluding the exotics? A reasonable expectation is that the metrics of Minkowski space M⁴ and CP₂fix completely the smooth structure of H= M⁴ x CP₂ but what about space-time surfaces X⁴ ⊆ H. The smooth structure, unlike topology, of X⁴ cannot be induced from that of H. In the case of Lie-groups, group structure implies the standard smooth structure: this is highly relevant for TGD.