What is meant by saying that any $1$-form (only on $S^3$?) can be uniquely written as a sum of a closed $1$-form and a "co-closed" $1$-form?
[...Since $H^1$ of $S^3$ is trivial it follows that the required closed $1$-form is also exact and then for any $1$-form say $A$ (a gauge field) on $S^3$ one says that it uniquely determines a gauge fixed 1-form $B$ such that $A = d\phi + B+\cdots$ ]