LOCALLY FLAT AND WILDLY EMBEDDED SEPARATRICES IN SIMPLEST MORSE-SMALE SYSTEMS

摘要

Let MS~(f low)(M~n,k) and MS~(diff)(M~n,k) be Morse-Smale flows and diffeomorphisms respectively the non-wandering set of those consists of A; 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 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,S) 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.