Morse–Smale systems with few non-wandering points

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 . For k = 3 , we show that the only values of n possible are n ∈ { 2 , 4 , 8 , 16 } , and M 2 is the projective plane. For n ⩾ 4 , M n is simply connected and orientable. We prove that the closure of any separatrix of f t ∈ MS flow ( M n , 3 ) is a locally flat n 2 -sphere while there is f t ∈ MS flow ( M n , 4 ) such that the closure of separatrix of f t is a wildly embedded codimension two sphere. This allows us to classify flows from MS flow ( M 4 , 3 ) . For n ⩾ 6 , one proves that the closure of any separatrix of f ∈ MS diff ( M n , 3 ) is a locally flat 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.