THE TOPOLOGY OF THE RELATIVE CHARACTER

VARIETIES

OF A QUADRUPLY-PUNCTURED SPHERE

ROBERT L. BENEDETTO AND WILLIAM M. GOLDMAN

Abstract. Let M be a quadruply-punctured sphere with boundary components A, B, C, D. The SL(2, ℂ)-character variety of M consists of equivalence classes of homomorphisms ρ of π_{1}(M) ⟶ SL(2, ℂ) and can be identified with a quartic hypersurface in ℂ^{7}. For fixed a, b, c, d ∈ ℂ, the subset V_{a,b,c,d} corresponding to representations ρ with tr(ρ(A)) = a, tr(ρ(B)) = b, tr(ρ(C)) = c, tr(ρ(D)) = d is a cubic surface in ℂ^{3}. We determine the singular points of V_{a,b,c,d} and classify its set V_{a,b,c,d}(ℝ) of ℝ-points into six topological types, at least when this set is nonsingular. V_{a,b,c,d}(ℝ) 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 SL(2, ℝ).