An extension of Lusternik-Schnirelmann category of closed 1-form to non compact manifolds
Michael Farber introduced the Lusternik-Schnirelmann category cat$(M,ξ)$ for the pair of finite CW complex $M$ and first-order cohomology $ξ$. 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 $ω$ on a closed manifold is less than cat$(M,[ω])$, then any gradient flows of $ω$ 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 $ω$ that occurs on non compact manifolds.