The topology of the relative character varieties of a quadruply-punctured sphere

Abstract

Let M be a quadruply-punctured sphere with boundary components A,B,C,D. The $\Slt$-character variety of M consists of equivalence classes of homomorphisms $\rho$ of $\pi_1(M)\longrightarrow\Slt$ and can be identified with a quartic hypersurface in $\funnyC^7$. For fixed $a,b,c,d\in\funnyC$, the subset $\va$ corresponding to representations $\rho$ with \thickmuskip 5mu plus 2mu minus 2mu $\tr(\rho(A)) = a$, $\,\tr(\rho(B)) = b$, $\,\tr(\rho(C)) = c$, $\,\tr(\rho(D)) = d$ is a cubic surface in $\funnyC^3$. We determine the singular points of $\va$ and classify its set $\Var$ of $\funnyR$-points into six topological types, at least when this set is nonsingular. $\Var$ contains a compact component if and only if $-2 < a,b,c,d < 2$. For certain values of (a,b,c,d), this component corresponds to representations in $\Slr$.