Topological characterization of canonical Thurston obstructions

TL;DR

This paper provides a topological characterization of canonical Thurston obstructions by generalizing Pilgrim's conjecture, showing that certain first-return maps lack obstructions, thus advancing understanding of Thurston maps.

Contribution

It proves a generalized form of Pilgrim's conjecture and offers a comprehensive topological characterization of canonical Thurston obstructions.

Findings

First-return maps of certain components are unobstructed

Complete topological characterization of canonical obstructions

Generalization of Pilgrim's conjecture

Abstract

Let f be an obstructed Thurston map with canonical obstruction \Gamma_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 \Gamma_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.