Topological characterization of canonical Thurston obstructions (Q1953854)

Let \(f\) be an orientation-preserving branched self-cover of the Riemann sphere \(S^2\) of degree \(d_f \geq 2\). A ``Thurston map'' is a pair \((f,Q_f)\) where \(f\) is such a mapping with finite post-critical set \(P(f)\) and \(Q_f\) is a finite set containing \(P(f)\). In [\textit{K. M. Pilgrim}, Adv. Math. 158, No. 2, 154--168 (2001; Zbl 1193.57002)], canonical Thurston obstructions were introduced and \textit{N. Selinger} showed in [Invent. Math. 189, No. 1, 111--142 (2012; Zbl 1298.37033)] that in the presence of a Thurston obstruction, there exists a canonical one. Continuing this work, in the paper under review, the author shows that if the first return map of a periodic component of the topological surface obtained from the sphere by pinching the curves of the canonical obstruction is a Thurston map, then the canonical obstruction of the first return map is empty. The author uses this result to give a complete topological description of canonical Thurston obstructions.