Theorem 1: Given a \( C^{\infty }\) compact manifold \( M\) furnished with a smooth volume form \( \mu \), there exists an \( F_{\sigma} \)-set containing all diffeomorphisms which are Cohomology Free in \( C^{\infty}_{\mu} ( M ) \).
The actual statement of this theorem is in fact way more constructive than the one that we just gave. For lack of space we do not give the alternative statement, which would reflect more accurately the content of its proof. The proof actually provides a good description of the \( F_{\sigma} \)-set in terms of the solvability of the cohomological equation, and the condition which is necessary for solvability is obtained in a constructive way (admitting the underlying non-constructivity of the problem and the notions involved).
Therefore, the term constructive is used in the following, quite loose, sense. When one has to prove the existence of a solution to such a difficult equation, the only reasonable way to do it is by constructing one. Thus, they take their screwdriver, go down to the street and, when they have gathered enough cash to live on for some months, come back to their desk and construct a solution using step-by-step approximation.
