Poincaré - Hopf Inequalities

In this article the main theorem establishes the necessity and sufficiency of the Poincar\'{e}-Hopf inequalities in order for the Morse inequalities to hold under the hypothesis that the flow and the reverse flow satisfy the Conley index duality condition on components of the chain recurrent set. The convex hull of the collection of all Betti number vectors which satisfy the Morse inequalities for a pre-assigned index data determines a Morse polytope defined on the nonnegative orthant. Using results from network flow theory, a scheme is provided for constructing all possible Betti number vectors which satisfy the Morse inequalities for a pre-assigned index data. Geometrical properties of this polytope are described.