Smooth 4-manifolds are important objects in low dimensional topology. This lecture series will introduce 4-manifolds from the following perspectives:
(1) classical invariants of 4-manifolds.
(2) Freedman's classification of simply-connected 4-manifolds (without proof).
(3) construction of 4-manifolds (Kirby calculus, surgery, rational blow-down).
(4) the Seiberg-Witten invariants and the Bauer-Furuta invariants of 4-manifolds.
(5) symplectic 4-manifolds.
(6) Donaldson's diagonalizability theorem.
(7) geography and botany problem of smooth 4-manifolds.
(8) exotic phenomena in dimension 4.
(9) embedded surfaces in 4-manifolds, the Thom conjecture and the Milnor conjecture.
(10) Khovanov homology and its application to 4-manifolds.(11) (time permitting) more recent developments (e.g. Gabai's light bulb theorem)
(1) Ronald Fintushel and Ronald Stern, “Six Lectures on 4-manifolds”
(2) John Morgan, “The Seiberg-Witten equations and Applications to the Topology of Smooth Four-manifolds”
(3) Simon Donaldson and Peter Kronheimer, “The Geometry of Four-Manifolds”
(4) Robert Gompf and Andras Stipsicz, “4-manifolds and Kirby Calculus”
(5) Lecture notes from Ciprian Manolescu's class “4-dimensional topology” at Stanford. (Notes written by Shintaro Fushida-Hardy)
Tencent Meeting ID：705 7478 7470
Wechat group: QR code
Office Hours: Wednesdays 2:00-3:00PM, 静斋309 or Tencent meeting (same ID)
Lecture 1: Why dimension 4 is special？
Lecture 2: Classical invariants of 4-manifolds
Lecture 3: Characteristic classes of 4-manifolds
Lecture 4: Complex surfaces as smooth 4-manifolds
Lecture 5: A crash course on Morse theory
Lecture 6: Kirby calculus I
Lecture 7: Kirby calculus II
Lecture 8: Trisection, rational blow down
Lecture 11: The Seiberg-Witten equations
Lecture 12: The Seiberg-Witten moduli space
Lecture 13: Properties of the Seiberg-Witten invariants
Lecture 14: Blow up formula and adjunction inequality
Lecture 15: The Thom conjecture and the Milnor conjecture
Lecture 17: Constructions of exotic smooth structures
Lecture 20: Khovanov homology and TQFT
Lecture 21: Lee homology and Rasmussen's s invariant
Lecture 22: Combinatorial proof of Milnor's conjecture