Isolating blocks for Morse flows

We present a constructive general procedure to build Morse flows on n-dimensional isolating blocks respecting given dynamical and homological boundary data recorded in abstract Lyapunov semi-graphs. Moreover, we prove a decomposition theorem for handles which, together with a special class of gluings, insures that this construction not only preserves the given ranks of the homology Conley indices, but it also is optimal in the sense that no other Morse flow can preserve this index with fewer singularities.