Basic forms and Morse–Novikov cohomology of Lie groupoids

Hao Ding

Mathematische Nachrichten, 2018, vol. 291, issue 13, 1989-2007

Abstract: Given a basic closed 1‐form on a Lie groupoid G, the Morse–Novikov cohomology groups Hθn(G) are defined in this paper. They coincide with the usual de Rham cohomology groups HdRn(G) when θ is exact and with the usual Morse–Novikov cohomology groups Hθn(M) when G 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.

Date: 2018
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