Classification Problems in Low-Dimensional Topology

Abstract

This thesis surveys classification problems and methods for manifolds in low-dimensional topology. We discuss examples of classification schemes in multiple equivalence relations. In Chapter 1, we introduce the intersection form and prove Whitehead's theorem, which classifies simply-connected, closed 4-manifolds up to homotopy equivalence. We also discuss complex surfaces and state the generalized Thom theorem, which establishes complex curves as genus-minimizers in simply connected closed complex surfaces. In Chapter 2, we introduce knot theory, discussing the slice-genus of torus knots, the Kronheimer–Mrowka adjunction inequality, and its resolution of the Milnor conjecture. We also present original results on the nonorientable slice genus of torus knots. In Chapter 3, we introduce Morse theory, Heegaard diagrams, and Kirby Calculus, culminating in the discussion of the classification of 3-manifolds arising from elementary surgery along torus knots in Chapter 4. In Chapter 5, we examine the classification by diffeomorphism type of a family of simply connected elliptic surfaces using logarithmic transforms and a proof of the Enriques–Kodaira classification in the simply-connected case. Finally, in Chapter 6, we discuss cobordisms as an equivalence relation and outline the proof of Smale's h-cobordism theorem, which yields a proof to the smooth Poincaré conjecture in dimensions five and above. We also examine why the Whitney trick fails in dimension four and how Freedman's use of Casson handles answers the topological 4-dimensional Poincaré conjecture in the affirmative.