Basic forms and Morse–Novikov cohomology of Lie groupoids

Abstract

Given a basic closed 1‐form on a Lie groupoid scriptG, the Morse–Novikov cohomology groups Hθnfalse(scriptGfalse) are defined in this paper. They coincide with the usual de Rham cohomology groups HdRnfalse(scriptGfalse) when θ is exact and with the usual Morse–Novikov cohomology groups Hθnfalse(Mfalse) when scriptG is the unit groupoid M⇉M generated by a smooth manifold M. We prove that the Morse–Novikov cohomology groups are invariant under Morita equivalences of Lie groupoids. On orbifold groupoids, we show that these groups are isomorphic to sheaf cohomology groups. Finally, when θ is not exact, we extend a vanishing theorem from smooth manifolds to orbifold groupoids.