In this paper we obtain relations between simple periodic surfaces of a vector field on a closed manifold M n , and the betti numbers of M n . When X is a gradient vector field of a nondegenerate function on M y the simple periodic surfaces are the critical points of the function and our relations are the Morse Inequalities. For Morse-Smale dynamical systems, the simple periodic surfaces are the critical points and closed orbits and we obtain the inequalities of Smale. In this case, where the

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... riodic surfaces are singularities and closed orbits, we are able to remove Smale's normal intersection condition and replace it by the much weaker condition: there are no cycles of orbits among the periodic surfaces. Consequently, in this context, the need for approximating gradient fields by Morse-Smale systems is eliminated. This is an announcement of the results; detailed proofs will appear elsewhere.