sphere


The Double Cap Conjecture ★★

Author(s): Kalai

Conjecture   The largest measure of a Lebesgue measurable subset of the unit sphere of $ \mathbb{R}^n $ containing no pair of orthogonal vectors is attained by two open caps of geodesic radius $ \pi/4 $ 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 $ 4 $-manifold has the homotopy type of the $ 4 $-sphere $ S^4 $, is it diffeomorphic to $ S^4 $?

Keywords: 4-manifold; poincare; sphere

Smooth 4-dimensional Schoenflies problem ★★★★

Author(s): Alexander

Problem   Let $ M $ be a $ 3 $-dimensional smooth submanifold of $ S^4 $, $ M $ diffeomorphic to $ S^3 $. By the Jordan-Brouwer separation theorem, $ M $ separates $ S^4 $ into the union of two compact connected $ 4 $-manifolds which share $ M $ as a common boundary. The Schoenflies problem asks, are these $ 4 $-manifolds diffeomorphic to $ D^4 $? ie: is $ M $ unknotted?

Keywords: 4-dimensional; Schoenflies; sphere

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