Topology of closed 1-forms and their critical points (Q5930181)

For any closed \(n\)-dimensional manifold \(X\) the author defines and studies the cup-length associated with the cohomology class \(\xi\in H^1(X; \mathbb{R})\) denoted by \(cl_k(\xi)\). It is defined by using the cohomological cup-products in flat bundles which are \(\xi\)-generic. The main result of this very interesting paper states that any closed 1-form \(\omega\) on \(X\) must have at least \(cl_k (\xi)-1\) geometrically distinct critical points, where \(\xi\) is the cohomology class of \(\omega\).NEWLINENEWLINENEWLINEAs the author mentions this theory opens nice perspectives to obtain infinite dimensional generalizations, some new estimations of the number of closed trajectories of Hamiltonian systems and possible applications in symplectic fixed points theory.