ec 2 00 2 Morse Novikov Theory and Cohomology with Forward Supports

We extend an approach to Morse theory due to Harvey and Lawson to non compact manifolds. This provides a modified and arguably more natural version of Morse-Novikov theory, where the Novikov Ring is replaced by a new " Forward Laurent Ring ". A detailed development of standard Morse-Novikov theory is clarified in several ways. Geometrically: The Novikov complex is described as a subcomplex of the DeRham complex of currents similar to [HL], although needing the important improvement by Minervini [M]. Topologically: The invariants are described in a new and concise way as " compact forward cohomology ". Algebraically: The Novikov Ring and the Forward Laurent ring are shown to be principal ideal domains, which are flat over the Laurent polynomials. Even for the Novikov ring these algebraic facts were not known before, though fundamental for the theory. Two dualities are established: first a back-ward/forward duality between infinite dimensional vector spaces over R; second a " Lambda duality " between finite dimensional vector spaces over the " Forward Laurent field " Λ. Suppose f : Y → R is a Morse-Smale function on an oriented (not necessarily compact) Riemannian manifold. Assume that the gradient vector field of f is complete, insuring a flow φ on Y. Recall the transversality condition of Smale: Smale For any two critical points p, q ∈ Cr (f), the stable manifold S p and the unstable manifold U q intersect transversally. To overcome lack of compactness of Y , we will also assume the following: Weakly Proper The function f is weakly proper if the intersection of each broken flow line with each slab f −1 ([a, b]) is compact.