Connecting Homomorphism and Separating Cycles

Roman V Ulvert,Siberian Federal University

doi:10.17516/1997-1397-2021-14-5-647-658

Roman V Ulvert, Siberian Federal University

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https://doi.org/10.17516/1997-1397-2021-14-5-647-658

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Abstract

We discuss the construction of a long semi-exact Mayer–Vietoris sequence for the homology of any finite union of open subspaces. This sequence is used to obtain topological conditions of representation of the integral of a meromorphic n-form on an n-dimensional complex manifold in terms of Grothendieck residues. For such a representation of the integral to exist, it is necessary that the cycle of integration separates the set of polar hypersurfaces of the form. The separation condition in a number of cases turns out to be a sufficient condition for representing the integral as a sum of residues. Earlier, when describing such cases (in the works of Tsikh, Yuzhakov, Ulvert, etc.), the key was the condition that the manifold be Stein. The main result of this article is the relaxation of this condition

Highlights

In the theory of functions of one complex variable the Cauchy residue of a function f at an isolated singular point a is represented in a local coordinate z by the integral over the cycle γ(a) = {|z − a| = ε} in a sufficiently small punctured neighborhood Ua \ {a}
We discuss the construction of a long semi-exact Mayer–Vietoris sequence for the homology of any finite union of open subspaces
It is possible to show that in order for the integral of a meromorphic form to be represented in terms of residues, it is necessary that the cycle of integration in a certain sense separates the set of polar hypersurfaces of the form

Summary

Introduction

The multidimensional analogue of the Cauchy residue is the Grothendieck residue of a meromorphic differential n-forms ω given on an n-dimensional complex-analytic manifold This residue in turn is represented by the integral over a local n-cycle in a neighborhood of an isolated intersection point of polar hypersurfaces of ω. In terms of the corresponding long Mayer–Vietoris sequence, we have obtained (Theorem 3.1) a necessary and sufficient condition under which any separating cycle is represented in terms of local cycles (and the integral is calculated in terms of the residues). This is only a reformulation of the problem in the language of homological algebra. The condition from Theorem 3.2 gives the desired relaxation of the condition of Steinnes of the manifold

Homology of the union of open subspaces

Connecting homomorphism

Separating cycles