Euler characteristic of spaces of real meromorphic functions (Q1812429)

A real meromorphic function is a pair \((f,\tau)\), where \(f: S_g\to\widehat{\mathbb C}\) is a ramified covering of genus \(g\) of the Riemann sphere and \(\tau: S_g\to S_g\) is an involution antiholomorphic with respect to the complex structure lifted from \(\widehat{\mathbb C}\) by the covering \(f\). There is a natural structure of a topological space on the set of real meromorphic functions. We denote this topological space by \(H\); it has countably many connected components. By a real decorated function we mean a quadruple \((f,\tau, E, \{D_e\})\), where \((f,\tau)\in H_0\) is a function with simple critical values, \(E\) is a finite subset of the sphere \(\widehat{\mathbb C}\), and \(\{D_e\}_{e\in E}\) is a set of pairwise disjoint closed disks in \(\widehat{\mathbb C}\). By the set of critical values of a decorated function \((f,\tau, E, \{D_e\})\) we mean the collection of points in \(\widehat{\mathbb C}\) consisting of the critical values of the function \((f,\tau)\) belonging to \(\widehat{\mathbb C}\setminus \bigcup_{e\in E}D_e\) and of the points \(e\in E\), where each \(e\) is endowed with the multiplicity equal to the number of critical values of the function \((f,\tau)\) belonging to \(D_e\). In this paper, for any connected component \(H_0\) of the space of real meromorphic functions, a compactification \(N(H_0)\) is constructed. The author proves that the space \(N_0\) is Hausdorff and compact, and defines an embedding of \(H_0\) into the space \(N(H_0)\). Let \(H_0^\ast\subset H_0\) be a submanifold consisting of all functions with simple critical values. The mapping \(C^\ast: H_0^\ast\to N(H_0)\) sends a function \((f,\tau)\) to the equivalent class of the decorated function \((f,\tau, \varnothing, \varnothing)\). The mapping \(C^\ast\) can be extended to a continuous embedding \(C: H_0\to N(H_0)\). Moreover, the image \(C(H_0)\) is open and dense in \(N(H_0)\). The Euler characteristic of the space \(H_0\), \(\chi(H_0)\), (of the space \(N(H_0)\), \(\chi(N(H_0))\),) is equal to the number of real meromorphic functions in \(H_0\) (to the number of equivalence classes of decorated functions in \(N(H_0)\), respectively) whose set of critical values consists of exactly two elements, \(i\) and \(-i\) (each having the multiplicity \(g + n- 1\)); \(\chi(H_0)\) and \(\chi(N(H_0))\) can be equal to either \(0\) or 1. Moreover, if \(\chi(H_0) = 1\), then the space \(H_0\) consists of functions of genus \(0\) (the converse assertion fails).