![](/files/happy5.png)
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.
![$ \mathbb{R}^n $](/files/tex/2010c953180b3521ec2f66d10e1f40ec71d44574.png)
![$ \pi/4 $](/files/tex/01608ea3b80f85b77096d16610a43e184782386c.png)
Keywords: combinatorial geometry; independent set; orthogonality; projective plane; sphere
Smooth 4-dimensional Poincare conjecture ★★★★
Author(s): Poincare; Smale; Stallings
Conjecture If a
-manifold has the homotopy type of the
-sphere
, is it diffeomorphic to
?
![$ 4 $](/files/tex/1f1498726bb4b7754ca36de46c0ccdd09136d115.png)
![$ 4 $](/files/tex/1f1498726bb4b7754ca36de46c0ccdd09136d115.png)
![$ S^4 $](/files/tex/8973308b8ba6ed78524b0e4751ab814bbaaa57e2.png)
![$ S^4 $](/files/tex/8973308b8ba6ed78524b0e4751ab814bbaaa57e2.png)
Keywords: 4-manifold; poincare; sphere
Smooth 4-dimensional Schoenflies problem ★★★★
Author(s): Alexander
Problem Let
be a
-dimensional smooth submanifold of
,
diffeomorphic to
. By the Jordan-Brouwer separation theorem,
separates
into the union of two compact connected
-manifolds which share
as a common boundary. The Schoenflies problem asks, are these
-manifolds diffeomorphic to
? ie: is
unknotted?
![$ M $](/files/tex/3f02401f624e31ef8679d3c3628c1f310058f388.png)
![$ 3 $](/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png)
![$ S^4 $](/files/tex/8973308b8ba6ed78524b0e4751ab814bbaaa57e2.png)
![$ M $](/files/tex/3f02401f624e31ef8679d3c3628c1f310058f388.png)
![$ S^3 $](/files/tex/02a9a17122cd1be0450f9ddf93c53e3feb250aad.png)
![$ M $](/files/tex/3f02401f624e31ef8679d3c3628c1f310058f388.png)
![$ S^4 $](/files/tex/8973308b8ba6ed78524b0e4751ab814bbaaa57e2.png)
![$ 4 $](/files/tex/1f1498726bb4b7754ca36de46c0ccdd09136d115.png)
![$ M $](/files/tex/3f02401f624e31ef8679d3c3628c1f310058f388.png)
![$ 4 $](/files/tex/1f1498726bb4b7754ca36de46c0ccdd09136d115.png)
![$ D^4 $](/files/tex/ae9c9010fd648efa84b8c7e9351f095414a07e92.png)
![$ M $](/files/tex/3f02401f624e31ef8679d3c3628c1f310058f388.png)
Keywords: 4-dimensional; Schoenflies; sphere
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