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A “nice” map colour theorem. (English) Zbl 1030.57037
The author considers certain triangulations associated to a meromorphic function $$f:M\to S^2={\mathbb C}\cup\{\infty\}$$, where $$M$$ is a closed and connected Riemann surface. Namely, for any such $$f$$ one can find a triangulation $$L$$ of $$S^2$$ whose vertex set contains all of the singular values of $$f$$ (i.e., those $$f(z)$$ such that $$f'(z)=0$$), and such that the pull-back $$K=f^{-1}(L)$$ of $$L$$ under $$f$$ is a simplicial complex that triangulates $$M$$. If $$L$$ has the minimal number $$4$$ of vertices, so that $$f$$ has at most $$4$$ singular values, then $$f$$ is called a minimal meromorphic function. The degree of $$f$$ is the cardinality $$|f^{-1}(w)|$$ of the inverse image of a non-singular point $$w$$ of $$S^2$$.
The main result (Theorem 1) of the paper asserts that such a minimal meromorphic function on a Riemann surface of genus $$g$$ has degree $$d\geq(\sqrt{g}+1)^2$$, and that conversely for any $$n\in\mathbb N$$ not divisible by $$2$$ or $$3$$, there exists a triangulation of a Riemann surface of genus $$g=(n-1)^2$$ and $$f$$-vector $$(4n, 6n^2, 4n^2)$$ which admits a minimal meromorphic function of degree $$d=n^2$$. This existence result is then extended to degree $$n^2+t$$ for all $$t\geq 2$$.
The key fact in the proof of Theorem 1 is that for minimal meromorphic functions $$f$$, the $$4$$-coloring of $$K=f^{-1}(L)$$ given by assigning color $$i$$ to each vertex of $$K$$ that maps to the $$i$$-th vertex of $$L=\partial\Delta^3$$ is nice: all $$4$$ colors appear in the closed star of each edge of $$K$$. The first part of Theorem 1 is proved as the special case $$t=4$$ of a more general lower bound on the number of vertices of a $$t$$-vertex-colorable triangulation of a surface of genus $$g$$. The existence proof proceeds by an explicit construction using Graeco-Latin squares, i.e., $$n\times n$$ matrices over $${\mathbb Z}/n \times {\mathbb Z}/n$$ such that all entries are distinct.
Two further results (re)proved in the paper are a holomorphic Reeb theorem (the number of singular values of $$f:M\to S^2$$ is never $$1$$, and if it is $$0$$ or $$2$$, then $$M=S^2$$), and a “nice map color theorem” (Theorem 4): Any closed oriented surface of genus $$g$$ can be triangulated by a “nicely” $$4$$-colored simplicial complex $$K$$, such that all vertices of one color class have valence $$3$$, and $$K$$ has degree $$d\geq 2g + \frac 12 (5+3\sqrt{1+8g})$$; this bound is best possible if $$g={t \choose 2}$$ with $$t\in\mathbb N$$. This construction uses a triangulation with $$1$$-skeleton $$K_{n,n,n}$$ of the surface of genus $${n-1\choose n-3}$$ due to G. Ringel and J. W. T. Youngs [Comment. Math. Helv. 45, 152-158 (1970; Zbl 0192.60507)]
Finally, in Theorem 5 the author provides a series of “nice” surfaces of genus $$g$$ and degree $$d=3(g+1)$$ for each $$g\geq 0$$ (the author notes that this is more than the bound of Theorem 4 for $$g>17$$). The $$g=2$$ member of this series had not been found by a computer search by F. Lutz based on random bistellar flips [see A. Björner and F. H. Lutz, Exp. Math. 9, 275-289 (2000; Zbl 1101.57306)].
MSC:
 57Q15 Triangulating manifolds 55M25 Degree, winding number 05C10 Planar graphs; geometric and topological aspects of graph theory 57N05 Topology of the Euclidean $$2$$-space, $$2$$-manifolds (MSC2010) 30F99 Riemann surfaces
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