N ov 2 00 8 CLOSED 1 - FORMS IN TOPOLOGY AND DYNAMICS

This article surveys recent progress of results in topology and dynamics based on techniques of closed one-forms. Our approach allows us to draw conclusions about properties of flows by studying ho-motopical and cohomological features of manifolds. More specifically we describe a Lusternik-Schnirelmann type theory for closed one-forms, the focusing effect for flows and the theory of Lyapunov one-forms. We also discuss recent results about cohomological treatment of the invari-ants cat(X, ξ) and cat 1 (X, ξ) and their explicit computation in certain examples. initiated a generalization of Morse theory which gives topological estimates on the numbers of zeros of closed 1-forms, see also [41, 42]. Novikov was motivated by a variety of important problems of mathematical physics, leading, in one way or another, to the problem of finding relations between the topology of the underlying manifold and the number of zeros which closed 1-forms in a specific one-dimensional real cohomology class possess. Recall that a closed 1-form can be viewed as a multi-valued function or functional whose branching behavior is fully characterized by the cohomology class. In [39] Novikov studies various problems of physics where the motion can be reduced to the principle of an extremal action S, which is a multi-valued functional on the space of curves, with the variation δS being a well-defined closed 1-form, cf. [38, 37, 40]. Among problems of this kind are the Kirchhoff equations for the motion of a rigid body in ideal fluid, the Leggett equation for the magnetic momentum, and others. His fundamental idea in [39] was based on a plan to construct a chain complex , now called the Novikov complex, which uses dynamics of the gradient flow in the abelian covering associated with the cohomology class. The dynamics of gradient flows appears traditionally in Morse theory providing a bridge between the critical set of a function and the global ambient topology.