Fundamental groups of the spaces of rational functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T18:10:15Z http://mathoverflow.net/feeds/question/13296 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/13296/fundamental-groups-of-the-spaces-of-rational-functions Fundamental groups of the spaces of rational functions algori 2010-01-28T22:59:34Z 2010-04-26T13:24:45Z <p>Here is a question which I asked myself (and couldn't answer) while reading "The topology of spaces of rational functions" by G. Segal.</p> <p>Let $X$ be a smooth complete complex curve (=a compact Riemann surface) of genus $g$ and let $Rat(X,d)$ be the space of all regular (=holomorphic) maps from $X$ to $\mathbf{P}^1(\mathbf{C})$ of degree $d$. In this question I'm interested in the fundamental group of the open subset $U(X,d)$ of $Rat(X,d)$ formed by all $f$ such that all critical points of $f$ are simple and all critical values are distinct. (A critical point is a point at which the derivative of $f$ vanishes; a critical value is the image of a critical point.) To be more specific, let's say I'd like to</p> <ol> <li><p>find a "nice" system of generators of $\pi_1(U(X,d))$;</p></li> <li><p>to describe, for each of these generators, its image under the map induced by the map $G$ from $U(X,d)$ to the configuration space $B(\mathbf{P}^1(\mathbf{C}),k)$ of unordered subsets of $\mathbf{P}^1(\mathbf{C})$ of cardinality $k=2(d+g-1)$ that takes $f$ to its branch divisor (i.e. the divisor of the critical points).</p></li> </ol> <p>Here are some remarks that may be useful (or may not):</p> <p>First, here is how one can think of the fundamental group of $Rat(X,d)$. By associating to every function its divisor of poles we get a map $F$ from $Rat(X,d)$ to the $d$-th symmetric power $S^d(X)$ of $X$.</p> <p>Assume $d> 2g-2$. By the Riemann-Roch theorem, for any degree $d$ divisor $D$ the linear space ${\cal{L}}(D)=H^0(X,{\cal{O}}(D))$ (which is formed by all rational functions $f$ such that for any $x\in X$ the order of the pole of $f$ at $x$ is at most the multiplicity of $x$ in $D$) is $d-g+1$. So $F$ is surjective and a fiber of $F$ is $\mathbf{C}^{d-g+1}$ minus some number of hyperplanes (these are given by the condition that order the pole of $f$ at a point $x$ of $D$ is less then the multiplicity of $x$ in $D$).</p> <p>The map $F$ is probably not a fibration. However, the fundamental group of $Rat(X,d)$ is spanned by the loops in a general fiber of $F$ going around one of the hyperplanes, and lifts of the loops in $S^d(X)$ (these are all of the form "one of the points moves along a loop in $X$ and the other stand still").</p> <p>Second, recall that the Jacobian $J(X)$ of $X$ is defined as follows. Integration along cycles gives an injective map $H_1(X,\mathbf{Z})\to\mathbf{C}^g=Hom(H^0(X,\Omega_X),\mathbf{C})$ and the Jacobian of $X$ is the quotient. Moreover, once we have chosen a base point $x$ in $X$, we get a natural map $j:X\to J(X)$ defined as follows: for any $x'\in X$ take a path $\gamma$ from $x$ to $x'$ and set $j(x')$ to be the image in $J(X)$ of the "integration along $\gamma$ function". This is well defined map that can be extended by $\mathbf{Z}$-linearity to $S^d(X)$.</p> <p>Abel's theorem says that two disjoint effective divisors are the divisors of the zeros and the poles of a rational function if and only if their images under $j$ coincide. This may be useful in this problem, but I don't see how.</p> http://mathoverflow.net/questions/13296/fundamental-groups-of-the-spaces-of-rational-functions/22594#22594 Answer by Nikos Apostolakis for Fundamental groups of the spaces of rational functions Nikos Apostolakis 2010-04-26T13:24:45Z 2010-04-26T13:24:45Z <p>From a topological point of view, rational functions are branched coverings of $S^2$. The fundamental group of the space of branched coverings is the group of "liftable braids". This group was calculated for $d=3$ by Birman and Wajnryb [1] and for $d=4$ by myself [2]. I have recently calculated the general case, the results should be published Any Time, Really Soon Now.</p> <ol> <li> Birman, Wanryb, <i>3--fold branched coverings and the mapping class group of a surface</i>, LNM 1167, 24-46 <li> Apostolakis, <i>On 4--fold covering moves</i>, Algebraic and Geometric Topology 3 (2003), 117-145. <li> Mullazzani, Piergallini, <i>Lifting Braids</i>, arXiv:math/0107117 </ol>