Meromorphic functions without real critical values and related braids (Q6166632)

Let \(\mathcal H_{d,g}\) denote the Hurwitz space of degree \(d\) meromorphic functions on genus Riemann surfaces. Motivated by the phenomenon of avoided level crossings of perturbations from mathematical physics, the authors study the open subset \(\mathcal H^{nr}_{g,d} \subset \mathcal H_{d,g}\) consisting of meromorphic functions with simple critical values, none of which is real. If \(f:E \to \mathbb C \mathbb P^1\) is a meromorphic function from \(\mathcal H^{nr}_{g,d}\), the pre-image \(N_f = f^{-1}(\mathbb R \mathbb P^1) \subset E\) is a disjoint union of smooth closed real curves \(O_1, \dots O_l\) with orientation given by the positive direction of \(\mathbb R \subset \mathbb R \mathbb P^1\) and the connected components of \(E \setminus N_f\) are separated into those mapping onto the positive and negative half planes, so that \(N_f\) is a separating oriented collection of ovals in the authors' terminology. Each oval \(O_j\) is assigned a positive integer \(m_j\) given by the degree of the map to \(\mathbb R \mathbb P^1 \cong S^1\) so that \(\sum m_j = d\). The key idea is to assign to \(f\) the isotopy class of the map \(S^1 \to (E^d/\Delta)/\mathrm{Sym}_d\) given by \(t \mapsto f^{-1} (t) \subset E\) considered in the Braid group \(\mathrm{Br}_d (E)\) on \(d\) strands. As seen in [\textit{J. S. Birman}, Braids, links, and mapping class groups. Based on lecture notes by James Cannon. Princeton, NJ: Princeton University Press (1975; Zbl 0305.57013)], this associates connected components of \(\mathcal H^{nr}_{g,d}\) to certain equivalence classes of braids. In the paper under review, the authors introduce subgroups of \textit{boundary braids} to which the braids of meromorphic functions naturally belong. They describe the boundary braids in terms of standard generators, viewing them in terms of configuration spaces. Furthermore, they describe the classes in the mapping class group \(\mathrm{Mod}(E)\) corresponding to the boundary braids via the Nielsen-Thurston classification [\textit{B. Farb} and \textit{D. Margalit}, A primer on mapping class groups. Princeton, NJ: Princeton University Press (2011; Zbl 1245.57002)]. Using their classification of the braids, the main theorem describes the connected components of \(\mathcal H^{nr}_{g,d}\) as the orbits of the mapping class group of a closed surface acting on the conjugacy classes of subgroups of boundary braids (when \(g=0\) or \(g=1\), the action is on the quotient of the braid group by the respective centers). As a consequence they recover several results of \textit{S. M. Natanzon} [Sov. Math., Dokl. 30, 724--726 (1984; Zbl 0599.14021); translation from Dokl. Akad. Nauk SSSR, 279, 803--805 (1984); \textit{S. M. Natanzon}, Sel. Math. 12, No. 3, 1 (1991; Zbl 0801.30034); translation from Tr. Semin. Vektorn. Tenzorn. Anal. 23, 79--103 (1988) and ibid. 24, 104--132 (1988)]. They apply their results to some special classes of meromorphic functions, such as those induced from generic projections of plane curves. The paper closes with some open problems.