Isolating Blocks for Periodic Orbits

In this article, we prove that the Lyapunov semi-graphs associated with 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 performing a round handle operation and considering a variety of situations. Since we are concerned in showing the existence of certain blocks, we keep the complexity of the manifolds in consideration under control by considering mainly manifolds with free homology groups, in particular, we consider connected sums of tori manifolds.