Symplectic Morse Theory: I

On manifolds with a specified closed differential form, we introduce a Morse-type cone complex with elements that can be generated by pairs of critical points of a Morse function. The differential of the complex consists of gradient flows and an integration of the closed form over spaces of gradient flow lines. We prove that the cohomology of this complex is independent of both the Riemannian metric and the Morse function used to define the complex. When the specified closed form is a power of the symplectic structure, we prove that the cohomology is isomorphic to the cohomology of differential forms of Tsai, Tseng and Yau (TTY). We also obtain Morse-type inequalities that bound the dimensions of the TTY cohomologies by the number of Morse critical points.Comment: 27 pages, 1 figur