We extend the Marcus-Schaeffer bijection between orientable rooted bipartitequadrangulations (equivalently: rooted maps) and orientable labeled one-facemaps to the case of all surfaces, that is orientable and non-orientable aswell. This general construction requires new ideas and is more delicate thanthe special orientable case, but it carries the same information. Inparticular, it leads to a uniform combinatorial interpretation of the countingexponent $\frac{5(h-1)}{2}$ for both orientable and non-orientable rootedconnected maps of Euler characteristic $2-2h$, and of the algebraicity of theirgenerating functions, similar to the one previously obtained in the orientablecase via the Marcus-Schaeffer bijection. It also shows that the renormalizationfactor $n^{1/4}$ for distances between vertices is universal for maps on allsurfaces: the renormalized profile and radius in a uniform random pointedbipartite quadrangulation on any fixed surface converge in distribution whenthe size $n$ tends to infinity. Finally, we extend the Miermont andAmbj{\o}rn-Budd bijections to the general setting of all surfaces. Ourconstruction opens the way to the study of Brownian surfaces for any compact2-dimensional manifold.
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机译：我们将Marcus-Schaeffer双向射影扩展到所有曲面的可定向有根二分图（等效为：根图）和可定向标记单面图之间，这也是可定向和不可定向的。这种一般的结构需要新的想法，并且比特殊的定向情况要精致得多，但是它承载着相同的信息。特别是，它导致欧拉特征为$ 2-2h $的可定向和不可定向的根连接图的计数指数$ \ frac {5（h-1）} {2} $以及它们生成的代数的统一组合解释功能类似于先前通过Marcus-Schaeffer双射在可定向情况下获得的功能。它还表明，顶点间距离的重归一化因子$ n ^ {1/4} $对于所有表面上的贴图都是通用的：当大小为$ n $时，在任何固定表面上均匀随机的有尖二分四边形的重归一化轮廓和半径在分布上收敛到无穷远。最后，我们将Miermont和Ambj {\ o} rn-Budd双射扩展到所有曲面的一般设置。我们的构造为研究任何二维二维流形的布朗面开辟了道路。
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