You are currently browsing the monthly archive for August 2012.

This morning I heard the awful news that Bill Thurston died last night. Many of us knew that Bill was very ill, but we all hoped (or imagined?) that he would still be with us for a while yet, and the suddenness of this is very harsh. As Sarah Koch put it in an email to me, “Although this was not unexpected, it is still shocking.” On the other hand, I am glad to hear that he was surrounded by family, and died peacefully.

I counted Bill as my friend, as well as my mentor, and I have many vivid and happy memories of time I spent with him. I hope that writing down a few of these reminiscences will be cathartic for me, and for others who are coping with this loss.

I am spending a few months in Göttingen as a Courant Distinguished Visiting Professor, and talking a bit to Laurent Bartholdi about rational functions — i.e. holomorphic maps from the Riemann sphere $\widehat{\mathbb C}$ to itself. A rational function is determined (up to multiplication by a constant) by its zeroes and poles, and can therefore generically be put in the form $f:z \to P(z)/Q(z)$ where P and Q are polynomials of degree $d$. If $d=1$ then $f$ is invertible, and is called a fractional linear transformation (or, sometimes, a Mobius transformation). The critical points are the zeroes of $P'Q-Q'P$; note that this is a polynomial of degree $\le 2d-2$ (not $2d-1$) and the images of these points under $f$ are the critical values. Again, generically, there will be $2d-2$ critical values; let’s call them $V$. Precomposing $f$ with a fractional linear transformation will not change the set of critical values.

The map $f$ cannot usually be recovered from $V$ (even up to precomposition with a fractional linear transformation); one needs to specify some extra global topological information. If we let $\overline{C}$ denote the preimage of $V$ under $f$, and let $C$ denote the subset consisting of critical points, then the restriction $f:\widehat{\mathbb C} - \overline{C} \to \widehat{\mathbb C} - V$ is a covering map of degree $d$, and to specify the rational map we must specify both $V$ and the topological data of this covering. Let’s assume for convenience that 0 is not a critical value. To specify the rational map is to give both $V$ and a representation $\rho:\pi_1(\widehat{\mathbb C} - V,0) \to S_d$ (here $S_d$ denotes the group of permutations of the set $\lbrace 1,2,\cdots,d\rbrace$) which describes how the branches of $f^{-1}$ are permuted by monodromy about $V$. Such a representation is not arbitrary, of course; first of all it must be irreducible (i.e. not conjugate into $S_e \times S_{d-e}$ for any $1\le e \le d-1$) so that the cover is connected. Second of all, the cover must be topologically a sphere. Let’s call the (branched) cover $\Sigma$ for the moment, before we know what it is. The Riemann-Hurwitz formula lets one compute the Euler characteristic of $\Sigma$ from the representation $\rho$. A nice presentation for $\pi_1(\widehat{\mathbb C}-V,0)$ has generators $e_i$ represented by small loops around the points $v_i \in V$, and the relation $\prod_{i=1}^{|V|} e_i = 1$. For each $e_i$ define $o_i$ to be the number of orbits of $\rho(e_i)$ on the set $\lbrace 1,2,\cdots,d\rbrace$. Then

$\chi(\Sigma) = d\chi(S^2) - \sum_i (d-o_i)$

If each $\rho(e_i)$ is a transposition (i.e. in the generic case), then $o_i=d-1$ and we recover the fact that $|V|=2d-2$.

This raises the following natural question:

Basic Question: Given a set of points $V$ in the Riemann sphere, and an irreducible representation $\rho:\pi_1(\widehat{\mathbb C} - V,0) \to S_d$ satisfying $\sum_i (d-o_i) = 2d-2$, what are the coefficients of the rational function $z \to P(z)/Q(z)$ that they determine (up to precomposition by a fractional linear transformation)?