Let S be a submanifold of M of real codimension 2, (using a metric on the normal bundle) I will construct an explicit closed smooth 2-form on M, call it A (a representative of the Poincare dual to S). Then I will prove that A-[S] = dT. Where S and T are: currents! (differential forms with distributional coefficients). In local coordinates if S is given by x=y=0, then [S] = D dxdy, where D is the Dirac delta in the real plane with mass 1 at the origin. T (‘global angular form'(connection)) is a smooth 1-form on the complement of S, it behaves like dt where x+iy= r exp(it). When we say dT we mean in the sense of currents, here we also have d^2=0 and we can talk about cohomology of currents.
A smooth differential form is a current in a natural way, as well as a submanifold (simplex) ( S goes to [S], or what is the same integrate over S)). de Rham proved that each of these maps induce an isomorphism in cohomology. The point of the computation in the first paragraph is to give an analysis free feeling of a corollary of this general theory. I might use greek letters during the talk.