- Banach-Tarski Paradox
- Paradoxical decompositions of groups
- "Squaring the circle"
- No paradox in the plane
- Handlebodies and Smales's h-cobordism Thm
- Poincare Conjecture
- Higher dimensions -- dim > 4
- Smale's proof
- Stalling's Engulfing Lemma
- dim = 4 -- Freedman
- dim = 3 -- Perelman
- Failed attempts
- Whitehead fails and discovers the Whitehead manifold
- Stalling's How-Not-to-Do-It
- Bing's near-miss
- special triangulations
- Shellability
- Unicoherent triangulations
- Papakyriakopoulos' strategies
- Rego-Rourke
- Dunwoody's try
- Morse Theory
- Teichmuller space
- Schoenflies Theorem
- Generalized Schoenflies Theorem
- Morton Brown's proof (using a Bing-type shrinking argument)
- Mazur-Morse proof, (using Mazur Swindle)
- Schoenflies in the plane -- elementary arguments
- Related: Jordan Separation --elementary arguments for the plane, polygonal curves
- Bing shrinking
- An Overview of the Eight 3-dimensional Geometries
- Ricci Flow
- Geometrization and Ricci Flow
- Special Solutions to Ricci Flow (Solitons, etc...)
- Maximum principles for tensors and use with Ricci Flow
- SnapPea
- Triangulation algorithm
- Uses of the cusp view
- Recognition of hyperbolic 3-manifolds
- The Hopf Fibration
- Orbifolds
- Knots in 3-manifolds
- Mapping Class Groups
- Fiber Bundles
- Vector bundles
- Surface bundles over the circle
- Surgery on 3-manifolds
- Lickorish Twist Theorem
- Lickorish-Wallace Theorem, e.g. orientable 3-manifold cobordism group is trivial
- Thurston's Hyperbolic Dehn Surgery
- Cobordism groups
- Wild/pathological objects
- Horned spheres and crumbled cubes
- Wild arcs -- Fox-Artin arcs
- The Smith Conjecture
- Double of an Alexander horned ball
- Relation to hyperbolization
- Relation to minimal surface theory
- Volume Conjectures
- Jones polynomial and volume
- Khovanov Homology and volume
- Jones Polynomial
- Kauffman State-Sum model
- Split links and diagrams
- Prime links and diagrams
- Colored Jones polynomials
- Khovanov Homology
- Characteristic Submanifolds
- Classification of Lens Spaces
- Haken manifolds
- Seifert Fiber spaces
- Regular Neighborhoods
- Existence and Uniqueness
- Marshall Cohen's simplicial neighborhood theorem
- Relative regular neighborhoods
- Zeeman's Conjecture
- Characteristic Classes
- Milnor's Exotic 7-spheres
- The Kirby Calculus
- 4-manifolds
- Scorecards
- Intersection forms
- Donaldson theory
- Exotic R^4s
- Non-triangulable 4-manifolds
- classification of simply connected 4-manifolds