Twisted Morse Cohomology and Lichnerowicz Cohomology

The focus of this chapter is on proving that the \(\eta \) -twisted -twisted Morse cohomology groups are isomorphic to the Lichnerowicz cohomology groups associated to \(-\eta \) . We define the twisted Morse-Smale-Witten cochain complex for a general bundle of abelian groups G, and then we restrict to the case where G is defined by a closed 1-form. In this case, the twisted Morse-Smale-Witten coboundary operator is the usual Morse-Smale-Witten coboundary operator with extra coefficients given by integrating the closed 1-form along the gradient flow lines. We discuss locally conformal symplectic locally conformal symplectic (LCS) manifolds locally conformal symplectic (LCS) and Lichnerowicz cohomology, and we prove twisted Morse theoretic versions of the Poincaré Lemma Poincaré Lemma and the de Rham Theorem de Rham Theorem. Relationships with sheaf cohomology sheaf cohomology are discussed.