An extension of Lusternik-Schnirelmann category of closed 1-form to non compact manifolds

Michael Farber introduced the Lusternik-Schnirelmann category cat$(M,\xi )$ for the pair of finite CW complex $M$ and first-order cohomology $\xi$. It is inspired by the Morse-Novikov theory, which is a closed 1-form version of the Morse theory. An important result of this theory is that if the number of zeros of a closed 1-form $\omega$ on a closed manifold is less than cat$(M,[\omega ])$, then any gradient flows of $\omega$ has at least one homoclinic cycle. This paper begins with an explanation of Lusternik-Schnirelmann theory of closed 1-form, and extends Farber’s results to general non-compact manifolds. We will also explain that Farber’s results hold equally well on non compact manifolds, and explain the new phenomena related gradient flows of $\omega$ that occurs on non compact manifolds.