Hurwitz spaces
by
Sergei Natanzon
MGU Moscow

A Hurwitz space H_{g, n} is a set of all analytic meromorphic function of genus g and degree n. That is the set of all pairs (P, f), where P is a compact Riemann surface of genus g and f:P® [`C]=CÈ¥ is a meromorphic function of degree n. Some natural subsets of H_{g, n} also are called Hurwitz spaces.

Let us (P, f) Î H_{g, n}. Points p Î P, where df=0, are called ramifications points. Its images are called branch points. There exists only finite number of branch points z Î [`C]. Type of z is (n₁, ..., n_k), where k is the number of preimages p₁, ..., p_k of z and n_i is the degree of f in a neighborhood of p_i. Branch points of type (2, 1, ..., 1) are called simple. An important class of Hurwits spaces is H_{g, n}(n₁, ..., n_k) consisting from f Î H_{g, n} such that ¥ is a branch point of the type (n₁, ..., n_k) and all finite branch points are simple.

Theorem 1.[1] Any space H_{g, n}(n₁, ..., n_k) is connected.

For n₁=¼ = n_k=1 this is a classical theorem of Hurwitz (1891). Recently Dubrovin proved that H_{g, n}(n₁, ..., n_k) has a natural structure of Frobenius manifold and thus describe a quantum cohomology of some manifold.

Consider now a more general type of Hurwitz spaces. We say that meromorphic functions (P₁, f₁) and (P₂, f₂) are topologically equivalent if there exist homeomorphisms j:P₁®P₂ and y:[`(C)]®[`(C)] such that yf₁=f₂j. A class t of the topological equivalence is called a topological type. Let H^t be the Hurwitz space of all meromorphic functions of a topological type t. It follows from Theorem 1 that any H_{g, n}(n₁, ..., n_k) is a space of type H^t. If f Î H^t and a Î Aut([`(C)]) @ PSL(2, C) then af Î H^t. Thus PSL(2, C) acts on H^t.

Theorem 2. [2]. Any space H^t/ PSL(2, C) is homeomorphic to R^m/Mod, where Mod is a discrete group.

For the space of Laurent's polynomials L_{n, m} it was proved by Arnold (1996) using deep theorems of theory of singularities. For the space H_{g, n}(1, ..., 1) it was proved in my paper (1986), using the theory of Fuchsian groups. The last method is good and for arbitrary H^t.

It is especially important compactifications of the Hurwitz spaces for mathematical and physical applications. Recently Turaev and the author proposed a purely topological compactification of H_{g, n}. It is constructed by decorated functions (f, E, {D_e}_{e Î E}) where f is a meromorphic function from H_{g, n}(1, ..., 1),    E Ì [`(C)] is a finite subset and D<sub>e</sub> Ì [`C] is a disk, containing e and more than 1 branch point. Decorated functions (f, E, {D_e}_{e Î E}) and (f¢, E¢, {D_e¢}_{e¢ Î E}) are equivalent if E¢=E and one go to another by a homotopy fixing E=E¢ and branch points outside of all D_e. . The compactification N_{g, n} of H_{g, n} is a sum of H_{g, n} and equivalent classes of decorated functions. The topology of H_{g, n} gives a topology on N_{g, n}.

Theorem 3 ([3]). N_{g, n} is compact and Hausdorff.

For a calculation of the Euler characteristic c(N_{g, n}) of N_{g, n} we consider regular labeled 2-coloured graphs. Such graph is a finite connected graph G such that every vertex of G is provided with one of two colors (red or blue) so that the endpoints of any edge have different colors. Every edge l Î G is provided with a positive integer n(l). Every vertex n Î G is provided with a non-negative integer g(n), where g(n)=0, if n is the end of only 1 edge l and also n(l)=1.

Let us define

n(G)= å l Î G  n(l),    g(G)= å n Î G  g(n)+1-c(G),

where c(G) is the Euler characteristic of G.

Let us consider the number G_{g, n} of all regular leveled 2-coloured graph G such that g(G)=g and n(G)=n and

n(G)+d(G)+ å n is red  (2g(n)-2)\geqslant 2,    n(G)+d(G)+ å n is blue  (2g(n)-2)\geqslant 2,

where d(G) is the number of edges of G.

Theorem 4 ([3]). c(N_{g, n})=G_{g, n}+4.

Reference

1. S.M.Natanzon. Topology of 2-dimensional coverings and meroporphic functions on real and complex algebraic curves, Sel. Math. Sov., 12:3 (1993), 251-291.

2. S.M.Natanzon. Spaces of meromorphic functions on Riemann surfaces, Amer. Math. Soc. Transl. (2) Vol.180 (1997), 175-180.

3. S.M.Natanzon, V.Turaev. A compactification of the Hurwitz space. Topology. (1999).

Date received: May 21, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacy-10.