  On a smooth manifold, we associate to any closed differential form a mapping
cone complex. The cohomology of this mapping cone complex can vary with the de
Rham cohomology class of the closed form. We present a novel Morse theoretical
description for the mapping cone cohomology. Specifically, we introduce a Morse
complex for the mapping cone complex which is generated by pairs of critical
points with the differential defined by gradient flows and an integration of
the closed form over spaces of gradient flow lines. We prove that the
cohomology of our cone Morse complex is isomorphic to the mapping cone
cohomology and hence independent of both the Riemannian metric and the Morse
function used to define the complex. We also obtain sharp inequalities that
bound the dimension of the mapping cone cohomology in terms of the number of
Morse critical points and the properties of the specified closed form. Our
results are widely applicable, especially for any manifold equipped with a
geometric structure described by a closed differential form. We also obtain a
bound on the difference between the number of Morse critical points and the
Betti numbers.
