  In 1997, Schaeffer described a bijection between Eulerian planar maps and
some trees. In this work we generalize his work to a bijection between
bicolorable maps on a surface of any fixed genus and some unicellular maps with
the same genus. An important step of this construction is to exhibit a
canonical orientation for maps, that allows to apply the same local opening
algorithm as Schaeffer.
  As an important byproduct, we obtain the first bijective proof of a result of
Bender and Canfield from 1991, when they proved that the generating series of
maps in higher genus is a rational function of the generating series of planar
maps.
