$Z$-oriented triangulations of surfaces
Abstract
The main objects of the paper are -oriented triangulations of connected closed -dimensional surfaces. A -orientation of a map is a minimal collection of zigzags which double covers the set of edges. We have two possibilities for an edge -- zigzags from the -orientation pass through this edge in different directions (type I) or in the same direction (type II). Then there are two types of faces in a triangulation: the first type is when two edges of the face are of type I and one edge is of type II and the second type is when all edges of the face are of type II. We investigate -oriented triangulations with all faces of the first type (in the general case, any -oriented triangulation can be shredded to a -oriented triangulation of such type). A zigzag is homogeneous if it contains precisely two edges of type I after any edge of type II. We give a topological characterization of the homogeneity of zigzags; in particular, we describe a one-to-one correspondence between -oriented triangulations with homogeneous zigzags and closed -cell embeddings of directed Eulerian graphs in surfaces. At the end, we give an application to one type of the -monodromy.
Cite
@article{arxiv.2001.02626,
title = {$Z$-oriented triangulations of surfaces},
author = {Adam Tyc},
journal= {arXiv preprint arXiv:2001.02626},
year = {2020}
}
Comments
One of the results of this preprint (Proposition 3) can be found in arXiv:1902.10788. Since I am a single author of this statement, I rewrite this result to make the material self-completed. The remaining two authors of arXiv:1902.10788 do not have any objections and do not plan to include this result in their forthcoming paper. arXiv admin note: substantial text overlap with arXiv:1902.10788