Isolating Blocks for Periodic Orbits

In this article we prove that the Lyapunov semi-graphs associated to periodic orbits are realizable by constructing isolating blocks $N^n$ for periodic orbits of Morse-Smale flows. We analyze the effects on the Betti numbers of a manifold after a round handle operation is performed and a variety of situations are considered. Since we are concerned in showing the existence of certain blocks we keep the complexity of the manifolds in consideration under control by considering essentially manifolds with free homology groups, in particular we consider connected sums of tori manifolds.