MATH6071A - Topics in Topology I - 2016/17

Textbooks

• Gompf & Stipsicz--4-manifolds and Kirby Calculus
• Scorpan--The wild world of 4-manifolds
• Kirby--The Topology of 4-manifolds
• Guillou & Marin--A la Recherche de la Topologie Perdue

References

• Freedman & Quinn--Topology of 4-manifolds
• Ozbagci & Stipsicz--Surgery on contact 3-manifolds and Stein surfaces
• Cieliebak & Eliashberg--From Stein to Weinstein and back

Pre-class Notes

• W1: [Scorpan] Chapters 1-2.
• W2: [Kirby] Chapter X.
• W3: [Gompf] Sec 9.1-2; [Guillou-Marin] Casson's Lecture 3; [Scorpan] Sec 4.2.
• W4: [Gompf] Sec 4.6; 6.1; [Guillou-Marin] Casson's Lecture 1; [Scorpan] Sec 2.5.
• W5: [Kirby] XII.3-4, XIV; [Gompf] 9.3-4; [Scorpan] Ch. 5.
• W7: [Scorpan] Ch.5;
• W8: [Gompf] 9.4
• W9: [Kirby] VI; [Scorpan] 11.4, 4.3;
• W10: [Kirby] IV; [Scorpan] 11.4, 10.7.
• W12:

Lecture Notes

• W1: Review of Smale's proof of the h-cobordism theorem in dim>5; Freedman's 4d TOP h-cobordism theorem; Casson handles
• W2: Wall's theorems (h-cobordisms; self-diffeomorphisms of 4-manifolds; stable equivalence of 4-manifolds)
• W3: Wall's h-cobordism theorem (strong form); Freedman's TOP version; non-product smooth h-cobordisms between 4-manifolds; Self-diffeomorphisms of 4-manifolds from embedded 2-spheres; the Gluck twist; Decomposing cobordisms; Akbulut's corks
• W4: Casson's embedding theorem (+ Freedman's theorem on Casson handles) and applications; kinky handles; plumbings and self-plumbings; Freedman's E8-manifold; higher dimensional exotic spheres from plumbings; Kirby diagram for plumbings and Casson handles; simplest Casson handle.
• W5: Cassson handles and the Whitehead continuum; large vs small exotic R^4's; comparisons with higher dimensions.
• W7: Quadratic forms realizable by smooth 4-maniolds; 11/8-conjecture and its siblings; Furuta's theorem; dissolvable 4-manifolds; Rokhlin's theorems on 3d- and 4d- oriented-/ spin- cobordism groups; the Rokhlin invariant and the Casson invariant; proof of Freedman's theorem on fake 4-balls (sketch).
• W8: Large exotic R^4's; classifications of exotic R^4's; other noncompact 4-manifolds
• W9: Arf invariants; Freedman-Kirby's generalization of Rokhlin's theorem; characteristic surfaces and w_2; characteristic cobordisms
• W10: The 4-d characteristic cobordism group; spin-structures for low-dimensional manifolds; low-dimensional spin-cobordism groups; relating spin-cobordism groups and characteristic cobordism groups; alternative formulation of Rokhlin-Freedman-Kirby theorem.
• W12: