Homology class of a Lagrangian Klein bottle
Abstract
It is shown that an embedded Lagrangian Klein bottle represents a nontrivial mod 2 homology class in a compact symplectic fourmanifold $(X,\omega)$ with $c_1(X)\cdot[\omega]>0$. (In versions 1 and 2, the last assumption was missing. A counterexample to this general claim and the first proof of the corrected result have been found by Vsevolod Shevchishin.) As a corollary one obtains that the Klein bottle does not admit a Lagrangian embedding into the standard symplectic fourspace.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2001
 arXiv:
 arXiv:math/0106122
 Bibcode:
 2001math......6122N
 Keywords:

 Mathematics  Symplectic Geometry;
 Mathematics  Complex Variables;
 Mathematics  Geometric Topology
 EPrint:
 Version 3  completely rewritten to correct a mistake