sphere
The Double Cap Conjecture ★★
Author(s): Kalai
Conjecture The largest measure of a Lebesgue measurable subset of the unit sphere of containing no pair of orthogonal vectors is attained by two open caps of geodesic radius around the north and south poles.
Keywords: combinatorial geometry; independent set; orthogonality; projective plane; sphere
Smooth 4-dimensional Poincare conjecture ★★★★
Author(s): Poincare; Smale; Stallings
Conjecture If a -manifold has the homotopy type of the -sphere , is it diffeomorphic to ?
Keywords: 4-manifold; poincare; sphere
Smooth 4-dimensional Schoenflies problem ★★★★
Author(s): Alexander
Problem Let be a -dimensional smooth submanifold of , diffeomorphic to . By the Jordan-Brouwer separation theorem, separates into the union of two compact connected -manifolds which share as a common boundary. The Schoenflies problem asks, are these -manifolds diffeomorphic to ? ie: is unknotted?
Keywords: 4-dimensional; Schoenflies; sphere