Negami's famous planar cover conjecture is equivalent to the statement that a connected graph can be embedded in the projective plane if and only if it has a projective planar cover. In 1999, Hliněný proposed extending this conjecture to higher genus non-orientable surfaces. In this paper, we put forward a natural extension that encompasses orientable surfaces as well; for every compact surface $Σ$, a connected graph $G$ has a finite cover embeddable in $Σ$ if and only if $G$ is embeddable in a surface covered by $Σ$.
As evidence toward this, we prove that for every surface $Σ$, the connected graphs with a finite cover embeddable in $Σ$ have bounded Euler genus. Moreover, we show that these extensions of Negami's conjecture are decidable for every compact surface of sufficiently large Euler genus, surpassing what is known for Negami's original conjecture. We also prove the natural analogue for countable graphs embeddable into a compact (orientable) surface. More precisely, we prove that a connected countable graph $G$ has a finite ply cover that embeds into a compact (orientable) surface if and only if $G$ embeds into a compact (orientable) surface.
Our most general theorem, from which these results are derived, is that there is a constant $c>0$ such that for every surface $Σ$, there exists a decreasing function $p_Σ:\mathbb{N} \to \mathbb{N}$ with $\lim_{g\to \infty}p_Σ(g) =0$ such that every finite cover embeddable in $Σ$ of any connected graph with Euler genus $g\ge c$ has ply at most $p_Σ(g)$.
Non-asphericity of strata of genus-one differentials and stability spaces
Boston College（波士顿学院）；The Ohio State University（俄亥俄州立大学）；Tsinghua University（清华大学）；Beijing Institute of Mathematical Sciences and Applications（北京数学科学与应用研究院）；Zhejiang University（浙江大学）
