In this lecture series we introduce the concepts of knot, link, 3-dimensional manifold, Alexander polynomial and so on, and prove some of their basic properties. We also discuss Morse and Floer theory. To help better understand the material, practice problems will be provided and their solutions collected.
The aim of the course is to build a strong foundation in low-dimensional topology. thus enabling students to start doing independent research in the field.
The main themes of the course are knot theory and Floer homology groups associated to 3-dimensional manifolds. Floer homology is currently one of the most advanced tools used in topology and related fields, with many possible applications. After covering the basics of low-dimensional topology, our main focus will be Heegaard Floer homology.
Knot, link, 3-dimensional manifold, Alexander polynomial, genus and fibredness, Morse theory, Floer homology
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |
Regular lecture course format with homework exercises
Course schedule | Required learning | |
---|---|---|
Class 1 | Knots, links, genus and fibredness. Formal properties of knot Floer homology. | to be specified in each lecture |
Class 2 | Three definitions of the Alexander polynomial (infinite cyclic cover, Rolfsen's surgical view, Seifert matrix). Seifert's theorem (on the genus and the degree of the Alexander polynomial). | to be specified in each lecture |
Class 3 | Neuwirth's theorem (on fiberedness and the leading coefficient of the Alexander polynomial), definition of the Alexander polynomial via Fox calculus. | to be specified in each lecture |
Class 4 | Kauffman’s state model、Conway's skein relation for the Alexander polynomial. Grid diagrams. | to be specified in each lecture |
Class 5 | Combinatorial definition of knot Floer homology, its grading and Euler characteristic. | to be specified in each lecture |
Class 6 | The (combinatorial) proof of d^2=0 and invariance. Outline of general Floer theory. | to be specified in each lecture |
Class 7 | Morse function, Morse lemma, changes in sublevel sets. Heegaard decomposition for 3-dimensional manifolds. | to be specified in each lecture |
Class 8 | Gradient flow, transversality, moduli spaces and their orientation. | to be specified in each lecture |
Class 9 | Compactness via broken flow lines, Morse complex, gluing. | to be specified in each lecture |
Class 10 | The isomorphism of Morse homology and singular homology. | to be specified in each lecture |
Class 11 | Symplectic geometry, Lagrangian submanifolds, action functional. | to be specified in each lecture |
Class 12 | Holomorphic curves, Lagrangian intersection homology, Maslov index. | to be specified in each lecture |
Class 13 | Heegaard diagrams, spin^c structures, Heegaard Floer homology for closed manifolds. | to be specified in each lecture |
Class 14 | d^2=0 and invariance. The original definition of knot Floer homology. | to be specified in each lecture |
Class 15 | Sutured Floer homology and the proof of genus and fibredness detection. | to be specified in each lecture |
none in particular
Some of the lecture will follow recent survey papers by Juhasz (arXiv:1310.3418) and Manolescu (http://arxiv.org/abs/1401.7107). For the Morse theory part, Hutchings's lecture notes (http://math.berkeley.edu/~hutching/teach/276-2010/mfp.ps) will be useful.
Evaluation will be based on homework. Details will be specified in the class.
Complex analysis (up to the Riemann mapping theorem), algebraic topology (homology and homotopy), and smooth manifolds (for example, integral trajectories of a vector field) will be useful. The most important thing, however, is to have an open and inquisitive mind!