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The relative cup-length in local Morse cohomology. (English) Zbl 07959961

Summary: 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.

MSC:

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
55N45 Products and intersections in homology and cohomology
58C05 Real-valued functions on manifolds

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