4-manifolds

Department: 
MATH
Course Number: 
8803-LAM
Hours - Lecture: 
3
Hours - Lab: 
0
Hours - Recitation: 
0
Hours - Total Credit: 
3
Typical Scheduling: 
Not regularly scheduled

Special Topics course offered in Spring 2018 by Peter Lambert-Cole on "4-Manifolds".

Prerequisites: 

MATH 6441

Course Text: 

"4-Manifolds and Kirby Calculus" by Gompf and Stipsicz

Topic Outline: 

Part 1: Algebraic topology of 4-manifolds:

Topics:

(co)homology, intersection forms, characteristic classes (Chern, Stiefel-Whitney, Pontryagin), spin structures, Rokhlin's Theorem, Freedman's Theorem

Part 2: Handle decompositions and Kirby calculus

Topics:

Handle decomposition, handle sliding and cancellation, embedded surfaces and their complements, branched covers, turning a manifold upside down

Part 3: Complex surfaces

Topics:

complete intersections in CP^n, elliptic surfaces, Enriques-Kodaira classification, geography of complex surfaces, Lefschetz pencils and Lefschetz fibrations

Part 4: Seiberg-Witten invariants and Exotic 4-manifolds

Topics:

exotic smooth structures, the SW equations, basic classes, torus surgery and blow-up formulas, Fintushel-Stern knot surgery, Donaldson's Diagonalization Theorem, Thom Conjecture