http://openproblemgarden.org/op/smooth_4_dimensional_schoenflies_problem Comments for "Smooth 4-dimensional Schoenflies problem" en http://openproblemgarden.org/op/smooth_4_dimensional_schoenflies_problem <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/alexander_j">Alexander</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/topology">Topology</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <div class="envtheorem"><b>Problem</b>&nbsp;&nbsp; Let <img class="teximage" src="/files/tex/3f02401f624e31ef8679d3c3628c1f310058f388.png" alt="$ M $" /> be a <img class="teximage" src="/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png" alt="$ 3 $" />-dimensional smooth submanifold of <img class="teximage" src="/files/tex/8973308b8ba6ed78524b0e4751ab814bbaaa57e2.png" alt="$ S^4 $" />, <img class="teximage" src="/files/tex/3f02401f624e31ef8679d3c3628c1f310058f388.png" alt="$ M $" /> diffeomorphic to <img class="teximage" src="/files/tex/02a9a17122cd1be0450f9ddf93c53e3feb250aad.png" alt="$ S^3 $" />. By the Jordan-Brouwer separation theorem, <img class="teximage" src="/files/tex/3f02401f624e31ef8679d3c3628c1f310058f388.png" alt="$ M $" /> separates <img class="teximage" src="/files/tex/8973308b8ba6ed78524b0e4751ab814bbaaa57e2.png" alt="$ S^4 $" /> into the union of two compact connected <img class="teximage" src="/files/tex/1f1498726bb4b7754ca36de46c0ccdd09136d115.png" alt="$ 4 $" />-manifolds which share <img class="teximage" src="/files/tex/3f02401f624e31ef8679d3c3628c1f310058f388.png" alt="$ M $" /> as a common boundary. The Schoenflies problem asks, are these <img class="teximage" src="/files/tex/1f1498726bb4b7754ca36de46c0ccdd09136d115.png" alt="$ 4 $" />-manifolds diffeomorphic to <img class="teximage" src="/files/tex/ae9c9010fd648efa84b8c7e9351f095414a07e92.png" alt="$ D^4 $" />? ie: is <img class="teximage" src="/files/tex/3f02401f624e31ef8679d3c3628c1f310058f388.png" alt="$ M $" /> unknotted? </div> </tr></td> </table> </td> </tr> </table> Alexander, J 4-dimensional Schoenflies sphere Topology http://openproblemgarden.org/op/smooth_4_dimensional_schoenflies_problem#comment Fri, 06 Nov 2009 19:11:52 +0100 rybu 37123 at http://openproblemgarden.org