  This article presents new bijections on planar maps. At first a bijection is
established between bipolar orientations on planar maps and specific
"transversal structures" on triangulations of the 4-gon with no separating
3-cycle, which are called irreducible triangulations. This bijection
specializes to a bijection between rooted non-separable maps and rooted
irreducible triangulations. This yields in turn a bijection between rooted
loopless maps and rooted triangulations, based on the observation that loopless
maps and triangulations are decomposed in a similar way into components that
are respectively non-separable maps and irreducible triangulations. This gives
another bijective proof (after Wormald's construction published in 1980) of the
fact that rooted loopless maps with $n$ edges are equinumerous to rooted
triangulations with $n$ inner vertices.
