Knot Theory and 4-Manifolds
Closed smooth 4-manifold topology contains many notoriously difficult problems. Some of those problems become rather more tractable when posed instead about smooth 4-manifolds with boundary. In this course I’ll discuss a few such problems (exotica, geometric simple connectivity, exotic surfaces) and introduce some tools from knot theory and 3-manifold topology which make these relative problems more approachable. We’ll keep an eye on developing tools in the relative setting which may someday help us understand the closed setting. The first lecture is intended to follow Stipshicz’s first lecture, and will survey the requisite handle calculus.
TAs for this course are Kyle Hayden (Columbia University) and Charles Stine (Brandeis)
Topological 4-manifolds: the disc embedding theorem and beyond
Abstract: In 1982 Freedman proved the 4-dimensional Poincare conjecture in the topological category. The key tool in the proof was the disc embedding theorem, which allows a topological version of the Whitney trick in certain cases in dimension four. Freedman’s work was significantly extended by Quinn, who established fundamental results in topological 4-manifolds, such as topological transversality, and smoothing results for noncompact 4-manifolds.
The geography problem and constructing exotic 4-manifolds
Abstract: This course will cover
1. Basics of 4-manifolds (intersection forms); the classification of intersection forms, Kirby diagrams and the fundamental theorem of Kirby calculus. The classification of smooth (simply connected) 4-manifolds up to homeomorphism. (From Chapter 1 and Chapter 4 of [GS])
2. Examples of 4-manifolds: complex surfaces and symplectic 4-manifolds Constructions: rational blow-down, Gluck transformation, logarithmic transformation, and maybe Luttinger surgery (Chapters 3 and 10 from [GS])
3. Short intro the Seiberg-Witten invariants, spin^c structures, the SW-function; the adjunction inequality and simple applications. The Thom conjecture (maybe) (Sections 1.4 and 2.4 from [GS]),
4. Exotic smooth structures on closed 4-manifolds (Section 8.5 of [GS]). Geography problems for complex, symplectic and smooth 4-manifolds.
5. Open problems, speculations. Some accessible problems. Some hard problems.
[GS] Gompf and Stipsicz “4-Manifolds and Kirby Calculus”
TAs for this course are Marco Marengon (Max Planck Institute, Bonn) and Stefan Mihajlovic (Alfréd Rényi Institute of Mathematics)