- Lecturer: Andras Juhasz
General Prerequisites:
B3.5 Topology and Groups (MT) and C3.1 Algebraic Topology (MT) are essential.
B3.2 Geometry of Surfaces (MT) and C3.3 Differentiable Manifolds (MT) are useful but not essential.
B3.2 Geometry of Surfaces (MT) and C3.3 Differentiable Manifolds (MT) are useful but not essential.
Course Term: Hilary
Course Lecture Information: 16 lectures
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.
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.
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, outline of proof and the Whitney trick. Application: The generalized Poincare 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, Dehn lemma, lens spaces, Dehn surgery, branched double cover
4-manifolds: Kirby calculus, the intersection form, Freedman’s and Donaldson’s theorems (without proof)
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, Dehn lemma, lens spaces, Dehn surgery, branched double cover
4-manifolds: Kirby calculus, the intersection form, Freedman’s and Donaldson’s theorems (without proof)