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International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 56, Pages 3539-3572
http://dx.doi.org/10.1155/S0161171203210115

Heegaard splittings and Morse-Smale flows

1Navigation Support Office, European Space Operations Center, Robert-Bosch Straße 5, Darmstadt D-64293, Germany
2Mathematics Department, University of Wisconsin, Madison, WI 53706, USA
3Department of Mathematics, ETH Zürich, Zürich 8092, Switzerland

Received 16 October 2002

Copyright © 2003 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We describe three theorems which summarize what survives in three dimensions of Smale's proof of the higher-dimensional Poincaré conjecture. The proofs require Smale's cancellation lemma and a lemma asserting the existence of a 2-gon. Such 2-gons are the analogues in dimension two of Whitney disks in higher dimensions. They are also embedded lunes; an (immersed) lune is an index-one connecting orbit in the Lagrangian Floer homology determined by two embedded loops in a 2-manifold.