# 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.

## 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