Locally flat and wildly embedded separatrices in simplest morse-smale systems

Abstract

Let MS flow (M n , k) and MS diff (M n , k) be Morse-Smale flows and diffeomorphisms respectively the non-wandering set of those consists of k fixed points on a closed n-manifold M n (n???4). We prove that the closure of any separatrix of f t ? MS flow (M n , 3) is a locally flat $ \frac{n}{2} $ -sphere while there is f t ? MS flow (M n , 4) the closure of separatrix of those is a wildly embedded codimension two sphere. For n???6, one proves that the closure of any separatrix of f ? MS diff (M n , 3) is a locally flat $ \frac{n}{2} $ -sphere while there is f ? MS diff (M 4, 3) such that the closure of any separatrix is a wildly embedded 2-sphere.