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)

Basic
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”

(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)

**Wechat group: QR code**

**Office Hours: Wednesdays 2:00-3:00PM, 静斋309 or Tencent meeting (same ID)**

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**Lecture Notes:**

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__