General Prerequisites: B3.5 Topology and Groups (MT) and C3.1 Algebraic Topology (MT) are essential. We will assume working knowledge of the fundamental group, covering spaces, homotopy, homology, and cohomology.
B3.2 Geometry of Surfaces (MT) and C3.3 Differentiable Manifolds (MT) are useful but not essential, though some prior knowledge of smooth manifolds and bundles should make the material more accessible.
Course Overview: Low-dimensional topology is the study of 3- and 4-manifolds and knots. The classification of manifolds in higher dimensions can be reduced to algebraic topology. These methods fail in dimensions 3 and 4. Dimension 3 is geometric in nature, and techniques from group theory have also been very successful. In dimension 4, gauge-theoretic techniques dominate.
This course provides an overview of the rich world of low-dimensional topology that draws on many areas of mathematics. We will explain why higher dimensions are in some sense easier to understand, and review some basic results in 3- and 4-manifold topology and knot theory.
Learning Outcomes: The students will become acquainted with topological and smooth manifolds. They will master important techniques from Morse theory and learn how to manipulate handle decompositions of manifolds. They will get an idea about the role of the h-cobordism theorem and the Whitney trick in higher-dimensional topology.
They will learn a variety of techniques in knot theory, including how to manipulate diagrams using Reidemeister moves, how to derive knot invariants from Seifert surfaces, and how some of these are related to 4-dimensional quantities. They will be able to represent 3-manifolds using Heegaard decompositions, how to write them as sums of prime pieces using normal surface theory, and how to construct 3-manifolds via Dehn surgery and branched double covers along links. Finally, they will be able to represent 4-manifolds using Kirby diagrams and how to determine their homeomorphism type using the intersection form.
Course Synopsis: The definition of topological and smooth manifolds. Morse theory, handle decompositions, surgery. Every group can be the fundamental group of a manifold in dimension greater than three. The h-cobordism theorem (without proof) and the Whitney trick. Application: The generalized Poincar\'{e} conjecture.
Knots and links: Reidemeister moves, Seifert surface and genus, Alexander polynomial, fibred knots, Jones polynomial, prime decomposition, 4-ball genus.
3-manifolds: Heegaard decompositions, unique prime decomposition, loop theorem (without proof), lens spaces, Dehn surgery, branched double cover.
4-manifolds: Kirby calculus, the intersection form, Freedman’s and Donaldson’s theorems (without proof).
