Morse–Smale systems with few non-wandering points

Abstract

Let MSflow(Mn,k) and MSdiff(Mn,k) be Morse–Smale flows and diffeomorphisms respectively the non-wandering set of those consists of k fixed points on a closed n-manifold Mn. For k=3, we show that the only values of n possible are n∈{2,4,8,16}, and M2 is the projective plane. For n⩾4, Mn is simply connected and orientable. We prove that the closure of any separatrix of ft∈MSflow(Mn,3) is a locally flat n2-sphere while there is ft∈MSflow(Mn,4) such that the closure of separatrix of ft is a wildly embedded codimension two sphere. This allows us to classify flows from MSflow(M4,3). For n⩾6, one proves that the closure of any separatrix of f∈MSdiff(Mn,3) is a locally flat n2-sphere while there is f∈MSdiff(M4,3) such that the closure of any separatrix is a wildly embedded 2-sphere.