Smooth four-dimensional Poincaré conjecture
A four-dimensional space can have the topology of a sphere. Must every smooth version be smoothly equivalent to the standard sphere?
Knots and manifolds
Smooth structures, curved spaces and the manifolds produced by knot surgery.
Subjects: Topology · Algebra and geometry
Problems in this collection
8 problemsBorel conjecture
For certain closed manifolds, does matching their homotopy structure force them to be the same topological shape?
Novikov conjecture
Do a manifold’s higher signatures stay unchanged when its homotopy structure is preserved?
Bott–Grove–Halperin conjecture
Does nonnegative sectional curvature force finite total rational homotopy in a closed simply connected manifold?
Blaschke conjecture
If shortest paths stay uniquely determined out to the full diameter, must the entire metric be one of the standard symmetric geometries?
Cabling conjecture
Can surgery on a knot create a genuinely decomposable 3-manifold only through the known cable-knot construction?
Purely cosmetic surgery conjecture
Can two distinct surgery slopes on the same nontrivial knot produce the same oriented three-manifold?
Volume conjecture
Does exponential growth of colored Jones evaluations recover the volume of every hyperbolic knot complement?