  We introduce the notion of fully simple maps, which are maps with non
self-intersecting disjoint boundaries. In contrast, maps where such a
restriction is not imposed are called ordinary. We study in detail the
combinatorics of fully simple maps with topology of a disk or a cylinder. We
show that the generating series of simple disks is given by the functional
inversion of the generating series of ordinary disks. We also obtain an elegant
formula for cylinders. These relations reproduce the relation between moments
and free cumulants established by Collins et al. math.OA/0606431, and implement
the symplectic transformation $x \leftrightarrow y$ on the spectral curve in
the context of topological recursion. We conjecture that the generating series
of fully simple maps are computed by the topological recursion after exchange
of $x$ and $y$. We propose an argument to prove this statement conditionally to
a mild version of symplectic invariance for the $1$-hermitian matrix model,
which is believed to be true but has not been proved yet.
  Our argument relies on an (unconditional) matrix model interpretation of
fully simple maps, via the formal hermitian matrix model with external field.
We also deduce a universal relation between generating series of fully simple
maps and of ordinary maps, which involves double monotone Hurwitz numbers. In
particular, (ordinary) maps without internal faces -- which are generated by
the Gaussian Unitary Ensemble -- and with boundary perimeters
$(\lambda_1,\ldots,\lambda_n)$ are strictly monotone double Hurwitz numbers
with ramifications $\lambda$ above $\infty$ and $(2,\ldots,2)$ above $0$.
Combining with a recent result of Dubrovin et al. math-ph/1612.02333, this
implies an ELSV-like formula for these Hurwitz numbers.
