Gradient flows with wildly embedded closures of separatrices (Q630206)

\textit{D. Pixton} constructed in [Topology 16, 167--172 (1977; Zbl 0355.58004)] a diffeomorphism on the three-dimensional sphere with a non-wandering set consisting of four fixed points, and a saddle such that the closure of the two-dimensional separatrix is a wildly embedded two-dimensional sphere. It is known from [\textit{W.-T. Wu} and \textit{G. Reeb}, Sur les espaces fibrés et les variétés feuilletées. Actualités scientifiques et industrielles. 1183. Publ. Inst. Math. Univ. Strasbourg. XI. Paris: Hermann \& Cie (1952; Zbl 0049.12602)] that if a flow on a compact manifold has exactly two fixed points then the manifold is a sphere. Moreover, in [Publ. Math., Inst. Hautes Étud. Sci. 14, 181--222 (1962; Zbl 0109.15701)], \textit{J. Eells} and \textit{N. H. Kuiper} proved that if a compact manifold has a Morse-Smale flow with exactly three fixed points, then the dimension of the manifold is either 2, 4, 8 or 16. In this paper, the authors prove that any compact manifold of dimension at least four has a Morse-Smale flow such that the closure of the separatrix of a saddle is a wildly embedded sphere of codimension 2. This is done by first constructing a Morse-Smale flow with four fixed points.