MiS Preprint Repository

The present paper contains an interpretation and generalization of Novikov\'s theory for Morse type inequalities for closed 1-forms in terms of concepts from Conley\'s theory for dynamical systems. We introduce the concept of a flow carrying a cocycle $\alpha$, (generalized) $\alpha$-flow for short, where $\alpha$ is a cocycle in bounded Alexander-Spanier cohomology theory. Gradient-like flows can then be characterized as flows carrying a trivial cocycle. We also define $\alpha$-Morse-Smale flows that allow the existence of "cycles" in contrast to the usual Morse-Smale flows. $\alpha$-flows without fixed points carry not only a cocycle, but a cohomology class, in the sense of [8], and we shall deduce a vanishing theorem for generalized Novikov numbers in that situation. By passing to a suitable cover of the underlying compact polyhedron adapted to the cocycle $\alpha$, we construct a so-called $\pi$-Morse decomposition for an $\alpha$-flow. On this basis, we can use the Conley index to derive generalized Novikov-Morse inequalitites, extending those of M. Farber [13]. In particular, these inequalities include both the classical Morse type inequalities (corresponding to the case when $\alpha$ is a coboundary) as well as the Novikov type inequalities ( when $\alpha$ is a nontrivial cocycle).
[8] R. Churchill, Invariant sets which carry cohomology. J. Diff. Equ. 13(1973), 523-550
[13] M. Farber, Conting zeros of closed 1-forms, Math. DG/9903133 23 Mar 1999