Topological characterization of canonical Thurston obstructions

Abstract

Let $f$ be an obstructed Thurston map with canonical obstruction$\Gamma_f$. We prove the following generalization of Pilgrim'sconjecture: if the first-return map $F$ of a periodic component $C$ ofthe topological surface obtained from the sphere by pinching thecurves of $\Gamma_f$ is a Thurston map then the canonical obstructionof $F$ is empty. Using this result, we give a complete topologicalcharacterization of canonical Thurston obstructions.