Bruhat numbers of a strong Morse function

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].