Bruhat Numbers of a Strong Morse Function
Petr Evgen'evich Pushkar', Misha Temkin · Doklady Mathematics · 2022
Let f be a Morse function on a manifold M such that all its critical values are pairwise distinct. Given such a function (together with a certain choice of orientations) and a field $$\mathbb{F}$$ , we construct a set of nonzero elements of the field, which are called Bruhat numbers. Under certain acyclicity conditions on M, the alternating product of all the Bruhat numbers does not depend on f (up to sign); thus, it is an invariant of the manifold. For any typical one-parameter family of functions on M, we provide a relation that links the Bruhat numbers of the boundary functions of the family with the number of bifurcations happening along a path in the family. This relation generalizes the result from [1].