We construct a pinching semiconjugacy from a quadratic polynomial fc(z) = z2 + c with c ∈ (0,1/4) to f1/4(z) = z2 + 1/4 on the sphere. By lifting this semiconjugacy to their natural extensions, we investigate the structure of the regular leaf space of f1/4 in detail. © 2003, Department of Mathematics, Tokyo Institute of Technology. All rights reserved.