Topological characterization of canonical Thurston obstructions

Let f be an obstructed Thurston map with canonical obstruction Γ_f. We prove the following generalization of Pilgrim's conjecture: if the first-return map F of a periodic component C of the topological surface obtained from the sphere by pinching the curves of Γ_f is a Thurston map then the canonical obstruction of F is empty. Using this result, we give a complete topological characterization of canonical Thurston obstructions.

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