The Generalized Long Mayer–Vietoris Sequence and Separating Cycles

The notion of a cycle separating a collection of hypersurfaces in a complex-analytic manifold appears in the theory of multidimensional residue in connection with the Grothendieck residue. Namely, the residue $\mathrm{res}_{a}\omega$ of the meromorphic $n$-form $\omega$ on an $n$-dimensional manifold is represented by the integral in which $n$-cycle of integration (the local cycle at the point $a$) in a certain sense separates the set of polar hypersurfaces of the form $\omega$. The most complete results on separating cycles were obtained by Tsikh and Yuzhakov (1975–1988) under the condition of Steinnes of the manifold. The use of the generalized long Mayer –Vietoris sequence allows us to replace this condition with more flexible conditions, described in the homology of intersections of elements of the open cover of the manifold associated with the family of hypersurfaces. As a result, we obtain sufficient conditions of representation of the integral of a meromorphic form in terms of Grothendieck residues.
Another application of the generalized long Mayer–Vietoris sequence is related to the calculation of the combinatorial coefficients involved in the Gelfond–Khovanskii formula for the global residue in an algebraic torus (2002). In this case, we consider $0$-dimensional cycles separating some connection of tropical hypersurfaces in $\mathbb{R}^n$.