A Study of Smooth Structures in the Topology of 4-Dimensional Manifolds

Abstract

The topology of 4-dimensional manifolds occupies a uniquely exceptional position in geometric and differential topology, primarily because dimension four is the only one that admits exotic smooth structures — a phenomenon absent in every other dimension. This study systematically investigates the smooth structural diversity of simply connected compact 4-manifolds through analysis of intersection forms, Seiberg-Witten invariants, Donaldson diagonalization, and knot surgery techniques. The primary objectives are to classify smooth structures using established topological invariants and to examine quantitative constraints on smooth structure existence imposed by Furuta's 10/8-inequality and the Fintushel-Stern formula. A deductive-analytical methodology is employed, drawing on verified theorems and tabulated data from the literature through 2021. Results demonstrate that manifolds homeomorphic but non-diffeomorphic to standard models proliferate via knot surgery constructions, particularly in the CP²#nCP² family. The Fintushel-Stern formula generates infinitely many exotic smooth structures indexed by Alexander polynomials of knots. These findings confirm the extreme complexity and incompleteness of smooth classification in dimension four, reinforcing the hypothesis that no finite algorithm can classify all simply connected smooth 4-manifolds.