Orientability versus pseudo-orientability

Topological orientability was first defined by Listing in 1861 followed closely by Möbius (1865); Möbius defined orientation by the existence of a coherent edges orientation of a net (polygonal) decomposition of a compact surface. After Euler's classification of compact surfaces, it was clear that the Möbious combinatorial definition only dependents on the surface and not on the particular polygonal decomposition. This makes orientability both a topological and a combinatorial property. A surface can be orientable or not, and so a given surface can support an orientable or a non-orientable map, but not both. In contrast, bi-orientable and non-bi-orientable maps (introduced by Steve Wilson in the seventies as pseudo-orientable maps) can both be present in the same surface. This makes bi-orientability a purely combinatorial property, nonetheless, it shows remarkable resemblances with orientability. These two categories of maps are connected by the Petrie dual operation.The present paper pretends to emphasise their similarities and their differences, having as main focus bi-orientable maps. In the present paper, maps are considered to have no semi-edges and no boundary.