Title: Introduction to 3-/4-manifolds
Speakers: Lok Cheng So (Monash University); Huizheng Guo (The George Washington University)
Dates: July 7–10, 2025
Venue: Room 201, Zhengxin Building
On-campus Contact: Xiao Wang, wangxiaotop@jlu.edu.cn
Detailed Schedule
July 7
09:30–11:30 | Lok Cheng So
Title: Hyperbolic geometry
Content: Poincaré disk model, hyperboloid model, upper half-space model; 2D/3D hyperbolic geometry, geodesics, arc length, volume, isometries (Möbius transformations); ideal boundary of hyperbolic 3-space, horospheres, isometric spheres
13:30–15:30 | Huizheng Guo
Title: Introduction to Algebraic Topology in 4 Dimensions
Content: Properties of algebraic invariants; Poincaré conjecture
July 8
09:30–11:30 | Lok Cheng So
Title: Hyperbolic knot theory
Content: Alternatingness, primeness, geometric structure, discrete isometry groups, hyperbolic manifolds, tetrahedra, gluing and completeness equations
13:30–15:30 | Huizheng Guo
Title: Kirby Calculus
Content: Handles and handle decomposition; handle moves (handle slides, handle cancellation); 4-dimensional aspects (dotted 1-circle notation, 1-handle attachment, Kirby diagrams); examples
July 9
09:30–11:30 | Lok Cheng So
Title: Finding hyperbolic structures for knot/link complements
Content: Thurston’s hyperbolization theorem, incompressible surfaces, checkerboard decomposition, chunk decomposition, Menasco (1984) on alternating links
13:30–15:30 | Huizheng Guo
Title: Generalized Lightbulb Theorem
Content: 3D version; 4D version (Gabai); boundary version; Dax invariants (disk version); extended relative Dax invariant on S¹-bundles over Σ
July 10
09:30–11:30 | Lok Cheng So
Title: Alternating links on surfaces and wrap-up
Content: Diagrams on surfaces (including nonorientable), virtual/twisted links
