A “nice” map colour theorem.

*(English)*Zbl 1030.57037The 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)].

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)].

Reviewer: Julian Pfeifle (Berlin)