András Stipsicz
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)
Combine lecture notes for all 5 lectures is available here: Stipsicz
Lecture 1
Lecture notes are available here Stipsicz1
Lecture 2
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Lecture 3
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Lecture 4
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Lecture 5
Lecture notes are available here Stipsicz 5