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Starting from the axiomatic description of meromorphic functions withprescribed analytic properties, we introduce the cosimplicial cohomology ofrestricted meromorphic functions defined on foliations of smooth complexmanifolds. Spaces for double chain-cochain complexes and coboundary operatorsare constructed. Multiplications of several restricted meromorphic functionswith non-commutative parameters, as well as for elements of double complexspaces are introduced and their properties are discussed. In particular, weprove that the construction of invariants of cosimplicial cohomology ofrestricted meromorphic functions is non-vanishing, independent of the choice ofthe transversal basis for a foliation, and invariant with respect to changes ofcoordinates on a smooth manifold and on transversal sections. As anapplication, we provide an example of general cohomological invariants, inparticular, generalizing the Godbillon--Vay invariant for codimension onefoliations.

Cosimplicial cohomology of restricted meromorphic functions on foliated manifolds