四维流形的拓扑

任课教师 Speaker:林剑锋
时间 Time: 每周三、五 9:50-11:25 2021-9-13 ~ 12-3
地点 Venue:西楼第三会议室

课程描述 Description

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)

预备知识 Prerequisites

Basic algebraic topology and differential topology.

参考资料 References

(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

Password:118118

Wechat group: QR code

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


 




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 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 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

Lecture 21: Lee homology and Rasmussen's s invariant

Lecture 22: Combinatorial proof of Milnor's conjecture