The relative cup-length in local Morse cohomology

Local Morse cohomology associates cohomology groups to isolating neighbourhoods of gradient flows of Morse functions on (generally non-compact) Riemannian manifolds $M$. We show that local Morse cohomology is a module over the cohomology of the isolating neighbourhood, which allows us to define a cup-length relative to the cohomology of the isolating neighbourhood that gives a lower bound on the number of critical points of functions on $M$ that are not necessarily Morse. Finally, we illustrate by an example that this lower bound can indeed be stronger than the lower bound given by the absolute cup-length.