Smooth 4-manifolds are important objects in low
dimensional topology. This lecture series will introduce 4-manifolds from the
(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
(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
permitting) more recent developments (e.g. Gabai's light bulb theorem)
algebraic topology and differential topology.
(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”
Simon Donaldson and Peter Kronheimer, “The Geometry of Four-Manifolds”
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)
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 9: The geography problem of irreducible 4-manifolds. Spin-c structures on 4-manifolds
Lecture 10: More on spin-c structures. Hodge decomposition theorem
Lecture 12: The Seiberg-Witten moduli space
Lecture 14: Blow up formula and adjunction inequality
Lecture 15: The Thom conjecture and the Milnor conjecture
Lecture 16: Proof of Donaldson's theorem and Furuta's theorem
Lecture 17: Constructions of exotic smooth structures
Lecture 18: Fintushel-Stern knot surgery conjecture, exotic surfaces
Lecture 19: Seiberg-Witten invariants of symplectic manifolds, Definition of Khovanov homology
Lecture 20: Khovanov homology and TQFT