We will cover important subjects in low-dimensional geometric topology, such as knots, links, three-manifolds, the Alexander polynomial, and Morse theory, all with a view toward developing Floer homology. Floer homology is central to modern topology and related fields, with many manifestations. In this course we concentrate on Heegaard Floer homology and its applications.
We aim to prepare students for research in low-dimensional topology.
knots, links, three-manifolds, 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
Course schedule | Required learning | |
---|---|---|
Class 1 | compactness via broken flow lines, gluing, Morse complex | Definitions and properties |
Class 2 | equivalence of Morse homology and singular homology | Definitions and properties |
Class 3 | symplectic geometry, Lagrangian submanifolds, action functional | Definitions and properties |
Class 4 | pseudoholomorphic curves, Lagrangian intersections, Maslov index | Definitions and properties |
Class 5 | Heegaard diagrams, spin^c structures | Definitions and properties |
Class 6 | Heegaard Floer homology of a closed three-manifold | Definitions and properties |
Class 7 | d^2=0, invariance, the original definition of knot Floer homology | Definitions and properties |
Class 8 | sutured Floer homology, genus and fibredness detection | Definitions and properties |
No textbook
Survey papers by Juhasz (arXiv:1310.3418) and Manolescu (http://arxiv.org/abs/1401.7107), plus online lecture notes by Hutchings (http://math.berkeley.edu/~hutching/teach/276-2010/mfp.ps).
based on homework assignments
Basic algebraic topology (homology, cohomology, and the fundamental group), complex analysis (Riemann mapping theorem), and the previous quarter of this class.
All questions are welcome.