On topology of manifolds admitting a gradient-like flow with a prescribed non-wandering set

We study relations between the structure of the set of equilibrium points of a gradient-like flow and the topology of the support manifold of dimension $4$ and higher. We introduce a class of manifolds that admit a generalized Heegaard splitting. We consider gradient-like flows such that the non-wandering set consists of exactly $\mu$ node and $\nu$ saddle equilibrium points of indices equal to either $1$ or $n-1$. We show that, for such a flow, there exists a generalized Heegaard splitting of the support manifold of genius $g=\frac{\nu-\mu+2}2$. We also suggest an algorithm for constructing gradient-like flows on closed manifolds of dimension $3$ and higher with prescribed numbers of node and saddle equilibrium points of prescribed indices.