The Morse-Smale Complex

The main aim of this paper is to present the construction of the Morse– Smale complex of a compact smooth manifold M with boundary and to establish the connection to the topology of M . This approach to connecting the analysis of an appropriate function f : M → R — respectively the dynamical system associated to such a function and a Riemannian metric g — with the topology of the manifold was introduced by Thom (see [21]) and Smale (see [17], [18] and [19]). In the more traditional approach developed by Morse (see [10] for an presentation of these ideas) the function f is used to construct a CW-space of the same homotopy-type as the manifold M . The approach presented here uses the unstable manifold of the negative gradient vector field of f with respect to g to construct a decomposition of M that enables one to extract topological information from it. We remark that the approach developed by Thom and Smale is often more suitable for studying infinite-dimensional manifolds such as loop-spaces than the more traditional approach. However, the techniques used to obtain similar results in the infinite-dimensional setting differ substantially from the techniques used in this paper. For an exposition of Morse Theory as a toy-model of infinite-dimensional issues, see [16], for instance. In the first chapter we start to introduce the basic terminology concerning Morse Theory and present a prove of the important Morse Lemma. Then we distinguish special pairs (f, g) we call them Morse–Smale pairs — of functions f : M → R and Riemannian metrices g. The main motivation is to gain control of the behaviour of the negative gradient vector field − gradg(f) near critical points. We also introduce some conditions that control the behaviour of f and g on the boundary ofM . The boundary conditions considered here are not the most common ones. The choice of these boundary conditions is motivated by the idea that the boundary should fit with the decomposition of M by the unstable manifolds. Consequently the Morse–Smale complex associated to the critical points on the boundary forms a sub-complex of the Morse–Smale complex of the whole manifold. Stable and unstable manifolds are introduced next and the Lyapunovproperty is established. We prove that the stable and unstable manifolds are sub-manifolds diffeomorphic to Euclidean spaces and state the Smale condition. Then we investigate if the conditions we imposed on the Morse–Smale pairs are generic. In order to do this we define jets and cite some facts concerning openness and density of certain subsets of smooth functions. The main result consists of two parts: First it is shown that the set of Morse functions is C∞-dense and C2-open in the set of all extensions of a given Morse-function