We extend the notion of graph homomorphism to cellularly embedded graphs (maps) by designing operations on vertices and edges that respect the surface topology; we thus obtain the first definition of map homomorphism that preserves both the combinatorial structure (as a graph homomorphism) and the topological structure of the surface (in particular, orientability and genus). Notions such as the core of a graph and the homomorphism order on cores are then extended to maps. We also develop a purely combinatorial framework for various topological features of a map such as the contractibility of closed walks, which in particular allows us to characterize map cores. We then show that the poset of map cores ordered by the existence of a homomorphism is connected and, in contrast to graph homomorphisms, does not contain any dense interval (so it is not universal for countable posets). Finally, we give examples of a pair of cores with an infinite number of cores between them, an infinite chain of gaps, and arbitrarily large antichains with a common homomorphic image.

翻译：我们将图同态的概念扩展到胞腔嵌入图（地图），通过设计尊重曲面拓扑结构的顶点和边操作，从而首次定义了同时保持组合结构（作为图同态）和曲面拓扑结构（特别是可定向性和亏格）的地图同态。诸如图的核及核上的同态序等概念随后被推广到地图。我们还为地图的各种拓扑特征（如闭路径的可收缩性）建立了纯组合框架，该框架特别允许我们刻画地图核。随后证明，由同态存在性排序的地图核偏序集是连通的，并且与图同态不同，它不包含任何稠密区间（因此对可数偏序集不具有普适性）。最后，我们给出了若干示例：一对核之间存在无穷多个核、无穷长的间隙链以及具有共同同态像的任意大反链。