This is the third in a series of papers studying the finite type invariants of
various w-knotted objects and their relationship to the Kashiwara-Vergne problem and
Drinfel'd associators. In this paper we present a
topological solution to the Kashiwara-Vergne problem. In particular we recover via a topological argument the
Alkeseev-Enriquez-Torossian~\cite{AlekseevEnriquezTorossian:ExplicitSolutions}
formula for explicit solutions of the Kashiwara-Vergne equations
in terms of associators.
We study a class of w-knotted objects: knottings of
{\em 2-dimensional foams} and various associated features in four-dimensioanl
space. We use a topological construction which we name the double tree construction
to show that every {\em expansion} (also known as {\em universal finite type
invariant}) of parenthesized braids extends first to an
expansion of knotted trivalent graphs (a well known result), and
then extends uniquely to an expansion of the w-knotted objects mentioned above.
In algebraic language, an expansion for parenthesized braids is
uniquely determined by {\em Drinfel'd associator} $\Phi$, and an expansion for
w-knotted objects is uniquely determined by a solution $V$ of the Kashiwara-Vergne
problem~\cite{KashiwaraVergne:Conjecture}, as reformulated by Alekseev
and Torossian~\cite{AlekseevTorossian:KashiwaraVergne}. Hence our
result provides a topological framework for the result of
\cite{AlekseevEnriquezTorossian:ExplicitSolutions}
that ``there is a formula for $V$ in terms of $\Phi$'', along with an
independent topological proof of the Kashiwara-Vergne Theorem and the Alekseev-Enriquez-Torossian formula.