ABSTRACT
We have seen that the cohomology Hk(M, C) on a manifold M is the quotient space
Hk(M, C) = ker { d : Γ
( M, EkM
)→ Γ (M, Ek+1M )} dΓ ( M, Ek−1M
) of the space of d-closed forms modulo the subspace of exact forms. So, a cohomology class is a set of forms {ω + dρ} for a fixed closed form ω and varying forms ρ. The purpose of the classical theory of De Rham-Hodge is to find, in a given cohomology class {ω + dρ}, a canonical representative ω0 which will be d-closed, and which will be specified by other equations. Moreover the cohomology class is 0, if and only if this canonical representation of the cohomology class is identically 0 as a form.
