Heegaard splittings and Morse-Smale flows
In this expository paper, the authors study the Heegaard-Morse-Smale (HMS) structures on smooth closed oriented connected 3-manifolds. The main objective of the given paper is to consider three theorems (Theorems 2.1, 2.2 and 2.3 in this paper) which explain the role of HMS structures in 3-dimensional topology and their relation to the Poincaré conjecture in dimension three. In the first part of the paper, the authors introduce the main objects and define the notions needed to state the three theorems. The remaining part of the paper contains the detailed analysis of the proof of these theorems. Theorem 2.1 establishes the existence of an HMS structure on every closed connected oriented 3-manifold. The proof uses Morse homology theory. Theorem 2.2 characterizes the class of integral homology 3-spheres via the existence of reduced HMS structures on such 3-manifolds. The proof also uses Morse homology theory and runs in the same way as in higher dimensions. Theorem 2.3 provides the necessary and sufficient conditions for a closed connected oriented 3-manifold to be diffeomorphic to the 3-sphere, in terms of the existence of numerically and geometrically reduced HMS structures and HMS structures of genus zero on it. The proof of the fact that the existence of zero genus HMS structure on a 3-manifold \(Y\) implies that this \(Y\) is diffeomorphic to the 3-sphere uses in an essential way the well-known result on the connectivity of the group Diff\(_+(S^2)\) [see also \textit{C. J. Earle} and \textit{J. Eells}, J. Differ. Geom. 3, 19--43 (1969; Zbl 0185.32901)]. The proof of the fact that a numerically reduced HMS structure on \(Y\) can be modified to the one having genus zero uses an improved form of Smale's cancellation lemma. One of the main counterparts in proving Theorem 2.3 is the assertion which claims that the existence of a geometrically reduced HMS structure on a 3-manifold \(Y\) implies the existence of a numerically reduced HMS structure on it. The proof of that fact uses the existence of a 2-gon which can be considered as an analogue in dimension two of Whitney discs in higher dimensions. An example of algebraically reduced structure on \(S^3\) which is not geometrically reduced is also given.
scientific article; zbMATH DE number 2012631